From dd5f9cb4424e071c9f8ad3958daea66d8cc6103d Mon Sep 17 00:00:00 2001 From: JyotsnaT Date: Tue, 22 Aug 2023 00:50:22 +0530 Subject: [PATCH 01/21] added 511 vectors --- answers/chapter5.md | 80 +++++++++++++++++++++++++++++++++++++++++++ answers/vectors1.png | Bin 0 -> 312910 bytes 2 files changed, 80 insertions(+) create mode 100644 answers/vectors1.png diff --git a/answers/chapter5.md b/answers/chapter5.md index e69de29b..bf56430a 100644 --- a/answers/chapter5.md +++ b/answers/chapter5.md @@ -0,0 +1,80 @@ +# 5.1.1 Vectors + +1. Dot product + 1. [E] What’s the geometric interpretation of the dot product of two vectors?\ + [A] In the geometric context, the dot product of 2 vecotrs is projection of one vector over another. + For the given geometric vectors $\overrightarrow A$ and $\overrightarrow B$, their dot product is given by\ + $\overrightarrow A.\overrightarrow B = |\overrightarrow A|.|\overrightarrow B|.cos\theta$ + ![](vectors1.png) + + 2. [E] Given a vector $u$, find vector $v$ of unit length such that the dot product of $u$ and $v$ is maximum.\ + [A] Dot product of $\vec{u}$ and $\vec{v}$ is givenby $|\vec{u}|.|\vec{v}|.cos\theta$. Where $|\vec{u}|$ and $\vec{v}$ are the magnitude of the vectors respectively. This dot product will be maximum when $cos\theta$ will be maximum, which occurs when $\theta = 0$ i.e both the vector overlap with each other.\ + Since, $max |\vec{u}|.|\vec{v}|.cos\theta$\ + $= |\vec{u}|.|\vec{v}|.1$\ + $= |\vec{u}|.1$\ + The vector $v$ of unit length to yield max dot product with $u$ is given by + $\vec{u}.\vec{v}$ = $= |\vec{u}|$\ + $\vec{v} = \frac{|\vec{u}|}{\vec{u}}$ + +2. Outer product + 1. [E] Given two vectors $a = [3, 2, 1]$ and $b = [-1, 0, 1]$. Calculate the outer product $a^Tb$?\ + [A] Given \ + $a = [3, 2, 1]$ $b = [-1, 0, 1]$\ + Their outer product is given as $a \times b$ which is also known as the 'cross product' given by $a^Tb$\ + $a^Tb = + \begin{bmatrix} + 3\\ + 2\\ + 1 + \end{bmatrix} \times + \begin{bmatrix} + -1 & 0 & 1 + \end{bmatrix} = + \begin{bmatrix} + -3 & 0 & 3\\ + -2 & 0 & 2\\ + -1 & 0 & 1 + \end{bmatrix} + $ + + 1. [M] Give an example of how the outer product can be useful in ML.\ + [A] Following are the use cases where the outer product of the vectors can be useful in ML. + 1. Measure orthogonality of 2 vectors: Two vectors are said to be orthogonal of the angle between them is $90\degree$ and the outer product among them is maximum. + 2. Correlation between 2 vectors + + +3. [E] What does it mean for two vectors to be linearly independent?\ + [A] Linearly independent vectors are orthogonal to each other. In such a situation, angel between the vectors will be $90\degree$. Which means their outer product will be zero.\ + $\vec{u}\times\vec{v} = |\vec{u}|.|\vec{v}|.sin\theta = 0$ + + +5. [M] Given two sets of vectors $A = {a_1, a_2, a_3, ..., a_n}$ and $B = {b_1, b_2, b_3, ... , b_m}$. How do you check that they share the same basis?\ + [A] In order to check whether two sets of vectors share the same basis, we need to find out the number of independent vectors in their respective vector spaces and compare if they are equal. The set of independent vectors is given by the rank of their augmented matrices. + + $A = {a_1, a_2, a_3, ..., a_n}$ can be written as \ + $A = + \begin{bmatrix} + a_1\\ + a_2\\ + a_3\\ + a_n + \end{bmatrix} + $ + Rank can be found by iteratively performing linear transformation among the rows till we achieve non zero rows those will be the linearly independent rows of the matrix and will be equal to the rank of this matrix. + +7. [M] Given $n$ vectors, each of $d$ dimensions. What is the dimension of their span?\ + [A] Dimension of the span of $n$ vectors is given by the rank of their augmented matrix. + + +9. Norms and metrics + 1. [E] What's a norm? What is $L_0, L_1, L_2, L_{norm}$?\ + [A] + 1. $L_0$ norm: Number of non zero elements in a vector. + 2. $L_1$ norm: Sum of absolute value of vector elements.\ + $|x|_1 = \sum{|x_i|}$ + 4. $L_2$ norm: Length of vector in Euclidean space.\ + $||x||_2 = (\sum{x_i^2})^{\frac{1}{2}}$ + 5. $L_{\infty}$ norm: Maximum absolute value of vector elemets.\ + $||x||_{\infty} = max{|x_i|}$ + 1. [M] How do norm and metric differ? Given a norm, make a metric. 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z6LYk2)_SsuMdynF#Y0}aWh!*yS6Fo6-D(=`w{0REv)@Kj-t}5He3|H8dRG}&tudgW z3I8|boRnjwC(-MM?;! zbyV~cx>YTegxeE6QrR`QDFEXrfH+5mvAQ$!MW1JlRq3IIy>ikOG_J?hkvh{n(YEW! zFQ#kzi|WxgG!wO@dA9ZEnTv=ULh@;4k}EuXdBq|OQDK##?HdmkTVe4rH03II1>9)D z0am|Uua=*Bsf$EwA>o6A+S=HAa?%q?(%!Oj(^>9+!I;!2Qe7qz-R_{w?Ydeti8sBfBt1x> zqy28~>p9Ovb+70w&I_taz{{IU65F>N3)2gJ(XW6Qw>9+%biSTvqkg@Po@7I$w=?zP zlMbzw&op#vWh5EH+8)!pwql%-jEIw#YNF_!FLh-}d%|S;WaH##brDRmBK1P*=MT89 zN*b1NGq!D$x_a!A8NEz*2tvS@pdOI%@9HZ)R`Fd_qpKOe2 z^R@x=2$wsHw#eRi6hkFR!g`0uI?Q0=OqfV`hH-}Oq~FNhks%C%vDgk#)hqmQXQC%G`Eg7OBxO zr{if8C#Be;Z9V1>Idt-yO7PF~w8dQMKY(7ul7_!5?hunOzby%~LnPVddrSAC^uu+% P@Xy>?_SQ*OZnXagq1}mD literal 0 HcmV?d00001 From d4d760c412971f5ffbf1bfd05a0781a0a49778bb Mon Sep 17 00:00:00 2001 From: JyotsnaT Date: Tue, 22 Aug 2023 00:56:42 +0530 Subject: [PATCH 02/21] fixing mathjax for 511 --- answers/chapter5.md | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/answers/chapter5.md b/answers/chapter5.md index bf56430a..41bc556e 100644 --- a/answers/chapter5.md +++ b/answers/chapter5.md @@ -4,7 +4,7 @@ 1. [E] What’s the geometric interpretation of the dot product of two vectors?\ [A] In the geometric context, the dot product of 2 vecotrs is projection of one vector over another. For the given geometric vectors $\overrightarrow A$ and $\overrightarrow B$, their dot product is given by\ - $\overrightarrow A.\overrightarrow B = |\overrightarrow A|.|\overrightarrow B|.cos\theta$ + $$\overrightarrow A.\overrightarrow B = |\overrightarrow A|.|\overrightarrow B|.cos\theta$$ ![](vectors1.png) 2. [E] Given a vector $u$, find vector $v$ of unit length such that the dot product of $u$ and $v$ is maximum.\ @@ -14,14 +14,14 @@ $= |\vec{u}|.1$\ The vector $v$ of unit length to yield max dot product with $u$ is given by $\vec{u}.\vec{v}$ = $= |\vec{u}|$\ - $\vec{v} = \frac{|\vec{u}|}{\vec{u}}$ + $$\vec{v} = \frac{|\vec{u}|}{\vec{u}}$$ 2. Outer product 1. [E] Given two vectors $a = [3, 2, 1]$ and $b = [-1, 0, 1]$. Calculate the outer product $a^Tb$?\ [A] Given \ - $a = [3, 2, 1]$ $b = [-1, 0, 1]$\ + $$a = [3, 2, 1], b = [-1, 0, 1]$$\ Their outer product is given as $a \times b$ which is also known as the 'cross product' given by $a^Tb$\ - $a^Tb = + $$a^Tb = \begin{bmatrix} 3\\ 2\\ @@ -35,7 +35,7 @@ -2 & 0 & 2\\ -1 & 0 & 1 \end{bmatrix} - $ + $$ 1. [M] Give an example of how the outer product can be useful in ML.\ [A] Following are the use cases where the outer product of the vectors can be useful in ML. @@ -45,21 +45,21 @@ 3. [E] What does it mean for two vectors to be linearly independent?\ [A] Linearly independent vectors are orthogonal to each other. In such a situation, angel between the vectors will be $90\degree$. Which means their outer product will be zero.\ - $\vec{u}\times\vec{v} = |\vec{u}|.|\vec{v}|.sin\theta = 0$ + $$\vec{u}\times\vec{v} = |\vec{u}|.|\vec{v}|.sin\theta = 0$$ 5. [M] Given two sets of vectors $A = {a_1, a_2, a_3, ..., a_n}$ and $B = {b_1, b_2, b_3, ... , b_m}$. How do you check that they share the same basis?\ [A] In order to check whether two sets of vectors share the same basis, we need to find out the number of independent vectors in their respective vector spaces and compare if they are equal. The set of independent vectors is given by the rank of their augmented matrices. $A = {a_1, a_2, a_3, ..., a_n}$ can be written as \ - $A = + $$A = \begin{bmatrix} a_1\\ a_2\\ a_3\\ a_n \end{bmatrix} - $ + $$ Rank can be found by iteratively performing linear transformation among the rows till we achieve non zero rows those will be the linearly independent rows of the matrix and will be equal to the rank of this matrix. 7. [M] Given $n$ vectors, each of $d$ dimensions. What is the dimension of their span?\ From 0bbc4cb39a96f0252b97cf039d05a2d989c46bb3 Mon Sep 17 00:00:00 2001 From: JyotsnaT Date: Tue, 22 Aug 2023 00:58:01 +0530 Subject: [PATCH 03/21] mathjax edit 2 511 --- answers/chapter5.md | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/answers/chapter5.md b/answers/chapter5.md index 41bc556e..9a550c43 100644 --- a/answers/chapter5.md +++ b/answers/chapter5.md @@ -23,16 +23,16 @@ Their outer product is given as $a \times b$ which is also known as the 'cross product' given by $a^Tb$\ $$a^Tb = \begin{bmatrix} - 3\\ - 2\\ + 3 \\ + 2 \\ 1 \end{bmatrix} \times \begin{bmatrix} -1 & 0 & 1 \end{bmatrix} = \begin{bmatrix} - -3 & 0 & 3\\ - -2 & 0 & 2\\ + -3 & 0 & 3 \\ + -2 & 0 & 2 \\ -1 & 0 & 1 \end{bmatrix} $$ @@ -54,9 +54,9 @@ $A = {a_1, a_2, a_3, ..., a_n}$ can be written as \ $$A = \begin{bmatrix} - a_1\\ - a_2\\ - a_3\\ + a_1 \\ + a_2 \\ + a_3 \\ a_n \end{bmatrix} $$ From 7e07636a0ed7c4e06992350ecf04f58ec5ef558d Mon Sep 17 00:00:00 2001 From: JyotsnaT Date: Tue, 22 Aug 2023 01:09:02 +0530 Subject: [PATCH 04/21] 511 update --- answers/chapter5.md | 25 ++++++++++++++----------- 1 file changed, 14 insertions(+), 11 deletions(-) diff --git a/answers/chapter5.md b/answers/chapter5.md index 9a550c43..bd04b56b 100644 --- a/answers/chapter5.md +++ b/answers/chapter5.md @@ -21,7 +21,8 @@ [A] Given \ $$a = [3, 2, 1], b = [-1, 0, 1]$$\ Their outer product is given as $a \times b$ which is also known as the 'cross product' given by $a^Tb$\ - $$a^Tb = + ```math + a^Tb = \begin{bmatrix} 3 \\ 2 \\ @@ -35,7 +36,7 @@ -2 & 0 & 2 \\ -1 & 0 & 1 \end{bmatrix} - $$ + ``` 1. [M] Give an example of how the outer product can be useful in ML.\ [A] Following are the use cases where the outer product of the vectors can be useful in ML. @@ -51,15 +52,17 @@ 5. [M] Given two sets of vectors $A = {a_1, a_2, a_3, ..., a_n}$ and $B = {b_1, b_2, b_3, ... , b_m}$. How do you check that they share the same basis?\ [A] In order to check whether two sets of vectors share the same basis, we need to find out the number of independent vectors in their respective vector spaces and compare if they are equal. The set of independent vectors is given by the rank of their augmented matrices. - $A = {a_1, a_2, a_3, ..., a_n}$ can be written as \ - $$A = - \begin{bmatrix} - a_1 \\ - a_2 \\ - a_3 \\ - a_n - \end{bmatrix} - $$ + $A = {a_1, a_2, a_3, ..., a_n}$ can be written as + ```math + A = +\begin{bmatrix} + a_1 \\ + a_2 \\ + a_3 \\ + a_n +\end{bmatrix} + + Rank can be found by iteratively performing linear transformation among the rows till we achieve non zero rows those will be the linearly independent rows of the matrix and will be equal to the rank of this matrix. 7. [M] Given $n$ vectors, each of $d$ dimensions. What is the dimension of their span?\ From 538c5dcb4194a89fbdca9d7bd6a72be5c1450f7d Mon Sep 17 00:00:00 2001 From: JyotsnaT Date: Tue, 22 Aug 2023 01:16:11 +0530 Subject: [PATCH 05/21] 511 update --- answers/chapter5.md | 11 +++++++---- 1 file changed, 7 insertions(+), 4 deletions(-) diff --git a/answers/chapter5.md b/answers/chapter5.md index bd04b56b..af03eb42 100644 --- a/answers/chapter5.md +++ b/answers/chapter5.md @@ -21,7 +21,7 @@ [A] Given \ $$a = [3, 2, 1], b = [-1, 0, 1]$$\ Their outer product is given as $a \times b$ which is also known as the 'cross product' given by $a^Tb$\ - ```math + $$ a^Tb = \begin{bmatrix} 3 \\ @@ -36,8 +36,11 @@ -2 & 0 & 2 \\ -1 & 0 & 1 \end{bmatrix} - ``` + $$ + ```math +\begin{bmatrix}X\\Y\end{bmatrix} + ``` 1. [M] Give an example of how the outer product can be useful in ML.\ [A] Following are the use cases where the outer product of the vectors can be useful in ML. 1. Measure orthogonality of 2 vectors: Two vectors are said to be orthogonal of the angle between them is $90\degree$ and the outer product among them is maximum. @@ -53,7 +56,7 @@ [A] In order to check whether two sets of vectors share the same basis, we need to find out the number of independent vectors in their respective vector spaces and compare if they are equal. The set of independent vectors is given by the rank of their augmented matrices. $A = {a_1, a_2, a_3, ..., a_n}$ can be written as - ```math + $$ A = \begin{bmatrix} a_1 \\ @@ -61,7 +64,7 @@ a_3 \\ a_n \end{bmatrix} - + $$ Rank can be found by iteratively performing linear transformation among the rows till we achieve non zero rows those will be the linearly independent rows of the matrix and will be equal to the rank of this matrix. From 566256b86cef05b9f7fd86a224d0a108bcb4eb25 Mon Sep 17 00:00:00 2001 From: JyotsnaT Date: Tue, 22 Aug 2023 01:17:39 +0530 Subject: [PATCH 06/21] update 511 --- answers/chapter5.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/answers/chapter5.md b/answers/chapter5.md index af03eb42..f13cfe8d 100644 --- a/answers/chapter5.md +++ b/answers/chapter5.md @@ -38,9 +38,9 @@ \end{bmatrix} $$ - ```math +```math \begin{bmatrix}X\\Y\end{bmatrix} - ``` +``` 1. [M] Give an example of how the outer product can be useful in ML.\ [A] Following are the use cases where the outer product of the vectors can be useful in ML. 1. Measure orthogonality of 2 vectors: Two vectors are said to be orthogonal of the angle between them is $90\degree$ and the outer product among them is maximum. From 48c8dbc2532410c63ac4cf3de6fb2efea14b2adb Mon Sep 17 00:00:00 2001 From: JyotsnaT Date: Tue, 22 Aug 2023 01:28:26 +0530 Subject: [PATCH 07/21] update 511 --- answers/chapter5.md | 20 ++++++-------------- 1 file changed, 6 insertions(+), 14 deletions(-) diff --git a/answers/chapter5.md b/answers/chapter5.md index f13cfe8d..e6b804d5 100644 --- a/answers/chapter5.md +++ b/answers/chapter5.md @@ -37,11 +37,8 @@ -1 & 0 & 1 \end{bmatrix} $$ - -```math -\begin{bmatrix}X\\Y\end{bmatrix} -``` - 1. [M] Give an example of how the outer product can be useful in ML.\ + + 2. [M] Give an example of how the outer product can be useful in ML.\ [A] Following are the use cases where the outer product of the vectors can be useful in ML. 1. Measure orthogonality of 2 vectors: Two vectors are said to be orthogonal of the angle between them is $90\degree$ and the outer product among them is maximum. 2. Correlation between 2 vectors @@ -56,15 +53,10 @@ [A] In order to check whether two sets of vectors share the same basis, we need to find out the number of independent vectors in their respective vector spaces and compare if they are equal. The set of independent vectors is given by the rank of their augmented matrices. $A = {a_1, a_2, a_3, ..., a_n}$ can be written as - $$ - A = -\begin{bmatrix} - a_1 \\ - a_2 \\ - a_3 \\ - a_n -\end{bmatrix} - $$ + + ```math +A = \begin{bmatrix}a_1\\a_2\\a_3\\a_n\end{bmatrix} + Rank can be found by iteratively performing linear transformation among the rows till we achieve non zero rows those will be the linearly independent rows of the matrix and will be equal to the rank of this matrix. From efaad6b37284dfdd05da1ae984b73fd9febe1303 Mon Sep 17 00:00:00 2001 From: JyotsnaT Date: Tue, 22 Aug 2023 01:29:22 +0530 Subject: [PATCH 08/21] update 511 --- answers/chapter5.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/answers/chapter5.md b/answers/chapter5.md index e6b804d5..33408e40 100644 --- a/answers/chapter5.md +++ b/answers/chapter5.md @@ -54,8 +54,9 @@ $A = {a_1, a_2, a_3, ..., a_n}$ can be written as - ```math +```math A = \begin{bmatrix}a_1\\a_2\\a_3\\a_n\end{bmatrix} +``` Rank can be found by iteratively performing linear transformation among the rows till we achieve non zero rows those will be the linearly independent rows of the matrix and will be equal to the rank of this matrix. From 2f7274aac24582c88f7ab2255f4e8277af6f5ef0 Mon Sep 17 00:00:00 2001 From: JyotsnaT Date: Tue, 22 Aug 2023 01:33:20 +0530 Subject: [PATCH 09/21] update 511 --- answers/chapter5.md | 23 ++++++----------------- 1 file changed, 6 insertions(+), 17 deletions(-) diff --git a/answers/chapter5.md b/answers/chapter5.md index 33408e40..df7f26e9 100644 --- a/answers/chapter5.md +++ b/answers/chapter5.md @@ -20,24 +20,13 @@ 1. [E] Given two vectors $a = [3, 2, 1]$ and $b = [-1, 0, 1]$. Calculate the outer product $a^Tb$?\ [A] Given \ $$a = [3, 2, 1], b = [-1, 0, 1]$$\ - Their outer product is given as $a \times b$ which is also known as the 'cross product' given by $a^Tb$\ - $$ - a^Tb = - \begin{bmatrix} - 3 \\ - 2 \\ - 1 - \end{bmatrix} \times - \begin{bmatrix} - -1 & 0 & 1 - \end{bmatrix} = - \begin{bmatrix} - -3 & 0 & 3 \\ - -2 & 0 & 2 \\ - -1 & 0 & 1 - \end{bmatrix} - $$ + Their outer product is given as $a \times b$ which is also known as the 'cross product' given by $a^Tb$ +```math +a^Tb = \begin{bmatrix}3\\2\\1\end{bmatrix} \times \begin{bmatrix}-1&0&1\end{bmatrix} = \begin{bmatrix}-3&0&3\\-2&0&2\\-1&0&1\end{bmatrix} +``` + +E 2. [M] Give an example of how the outer product can be useful in ML.\ [A] Following are the use cases where the outer product of the vectors can be useful in ML. 1. Measure orthogonality of 2 vectors: Two vectors are said to be orthogonal of the angle between them is $90\degree$ and the outer product among them is maximum. From 85d8ad8e8f9b7219df038d568be375a9b285dbfa Mon Sep 17 00:00:00 2001 From: JyotsnaT Date: Wed, 30 Aug 2023 21:39:30 +0530 Subject: [PATCH 10/21] matrices solutions 1 --- answers/chapter5.md | 89 ++++++++++++++++++++++++++++++++++++++++++++- 1 file changed, 87 insertions(+), 2 deletions(-) diff --git a/answers/chapter5.md b/answers/chapter5.md index df7f26e9..704d20f4 100644 --- a/answers/chapter5.md +++ b/answers/chapter5.md @@ -26,7 +26,7 @@ a^Tb = \begin{bmatrix}3\\2\\1\end{bmatrix} \times \begin{bmatrix}-1&0&1\end{bmatrix} = \begin{bmatrix}-3&0&3\\-2&0&2\\-1&0&1\end{bmatrix} ``` -E + 2. [M] Give an example of how the outer product can be useful in ML.\ [A] Following are the use cases where the outer product of the vectors can be useful in ML. 1. Measure orthogonality of 2 vectors: Two vectors are said to be orthogonal of the angle between them is $90\degree$ and the outer product among them is maximum. @@ -47,7 +47,6 @@ E A = \begin{bmatrix}a_1\\a_2\\a_3\\a_n\end{bmatrix} ``` - Rank can be found by iteratively performing linear transformation among the rows till we achieve non zero rows those will be the linearly independent rows of the matrix and will be equal to the rank of this matrix. 7. [M] Given $n$ vectors, each of $d$ dimensions. What is the dimension of their span?\ @@ -66,3 +65,89 @@ A = \begin{bmatrix}a_1\\a_2\\a_3\\a_n\end{bmatrix} $||x||_{\infty} = max{|x_i|}$ 1. [M] How do norm and metric differ? Given a norm, make a metric. Given a metric, can we make a norm?\ [A] + + + +# 5.1.2 Matrices + +1. [E] Why do we say that matrices are linear transformations?\ + [A] A linear transformation is a function $g: R^m \rightarrow R^n$ that maps a vector space of dimension $m$ to another vector space of dimension $n$. A linear transformation satisfies following propoerties + 1. Homogenous\ + $g(cx) = cg(x)$ + 3. Additive\ + $g(x+y) = g(x) + g(y)$ + + When a matrix of dimension $A_{n\times m}$, is multiplied with a vector $v$ of dimension $n$ it results in another vector $u$ of dimension $m$, which inherently a transformation process. + $$A_{n\times m}v = u$$ + This transformation via matrices follows the above properties of linearity. Hence matrices are known as linear transformations. + +2. [E] What’s the inverse of a matrix? Do all matrices have an inverse? Is the inverse of a matrix always unique?\ + [A] Inverse of a matrix $A$ is another matrix $A^{-1}$ such that when multiplied together it yields an identity matrix. + $$AA^{-1} = I$$ + Identity matrix can be obtained by + $$A^{-1} = \frac{1}{|A|}Adj(A)$$ + Where $|A|$ is the determinant and $\frac{1}{|A|}Adj(A)$ is the adjoint. For the matrices where the determinant is $0$ the inverse is not defined. Such matrices are also known as *singular matrices*.\ + Inverse of a matrix is always unique. +3. [E] What does the determinant of a matrix represent?\ + [A] The determinant of the matrix is given by + $$det(A) = |A| = \sum_{i=1}^{n}A_{ij}C_{ij}$$ where $C_{ij}$ is the cofactor matrix. Determinant is only defined for square matrices. It represents the area/volume enclosed between the vectors. + +5. [E] What happens to the determinant of a matrix if we multiply one of its rows by a scalar $t \times R$?\ + [A] As per the row scaling property, on multiplying one of the rows with a scalar $t$ the resulting determinant will be scaled by $t$. + $$det(t\times A) = t\times det(A)$$ + +6. [M] A $4 \times 4$ matrix has four eigenvalues $3, 3, 2, -1$. What can we say about the trace and the determinant of this matrix?\ + [A] Trace of a matrix is given by the sum of it's eigen values.\ + Trace of the matrix : For a diagonal matrix, the sum of all it's diagonal elements. + $$Tr(A) = \sum_{i=1}^{n}a_{ij}$$ + Eigen values and eigen vectors: For the given relation + $$AX = \lambda X$$ + $\lambda$ is the eigen value of $A$ and $X$ is eigen vector of A given $X != 0$. The charecteristic equation of $A$ is given by + $$det(A-\lambda I)=0$$ whose roots give the eigen values of the Matrix $A$ as $\lambda _1$, $\lambda _2$, .. $\lambda _n$ + +8. [M] Given the following matrix:
+ $$ + \begin{bmatrix} + 1 & 4 & -2 \\ + -1 & 3 & 2 \\ + 3 & 5 & -6 + \end{bmatrix} + $$ + + Without explicitly using the equation for calculating determinants, what can we say about this matrix’s determinant? + **Hint**: rely on a property of this matrix to determine its determinant. + [A] The determinant of this matrix is $0$. + We can perform some linear transformation to the columns and observe one of the columns can become all $0$s which will make the determinant to be $0$. + $$ + \begin{bmatrix} + 1 & 4 & -2 \\ + -1 & 3 & 2 \\ + 3 & 5 & -6 + \end{bmatrix} = -2\times + \begin{bmatrix} + 1 & 4 & 1 \\ + -1 & 3 & -1 \\ + 3 & 5 & 3 + \end{bmatrix} = \begin{bmatrix} + 1 & 4 & 0 \\ + -1 & 3 & 0 \\ + 3 & 5 & 0 + \end{bmatrix} + $$ + +10. [M] What’s the difference between the covariance matrix $A^TA$ and the Gram matrix $AA^T$? + +12. Given $A \in R^{n \times m}$ and $b \in R^n$ + 1. [M] Find $x$ such that: $Ax = b$. + 1. [E] When does this have a unique solution? + 1. [M] Why is it when A has more columns than rows, $Ax = b$ has multiple solutions? + 1. [M] Given a matrix A with no inverse. How would you solve the equation $Ax = b$? What is the pseudoinverse and how to calculate it? + +13. Derivative is the backbone of gradient descent. + 1. [E] What does derivative represent? + 1. [M] What’s the difference between derivative, gradient, and Jacobian? + +14. [H] Say we have the weights $w \in R^{d \times m}$ and a mini-batch $x$ of $n$ elements, each element is of the shape $1 \times d$ so that $x \in R^{n \times d}$. We have the output $y = f(x; w) = xw$. What’s the dimension of the Jacobian $\frac{\delta y}{\delta x}$? +15. [H] Given a very large symmetric matrix A that doesn’t fit in memory, say $A \in R^{1M \times 1M}$ and a function $f$ that can quickly compute $f(x) = Ax$ for $x \in R^{1M}$. Find the unit vector $x$ so that $x^TAx$ is minimal. + + **Hint**: Can you frame it as an optimization problem and use gradient descent to find an approximate solution? \ No newline at end of file From db07b44b5ae676e5bf5be650a4b51af439164426 Mon Sep 17 00:00:00 2001 From: JyotsnaT Date: Wed, 30 Aug 2023 21:58:21 +0530 Subject: [PATCH 11/21] 512 remove spaces matrix --- answers/chapter5.md | 32 +++++++------------------------- 1 file changed, 7 insertions(+), 25 deletions(-) diff --git a/answers/chapter5.md b/answers/chapter5.md index 704d20f4..2b493b86 100644 --- a/answers/chapter5.md +++ b/answers/chapter5.md @@ -26,7 +26,7 @@ a^Tb = \begin{bmatrix}3\\2\\1\end{bmatrix} \times \begin{bmatrix}-1&0&1\end{bmatrix} = \begin{bmatrix}-3&0&3\\-2&0&2\\-1&0&1\end{bmatrix} ``` - +a 2. [M] Give an example of how the outer product can be useful in ML.\ [A] Following are the use cases where the outer product of the vectors can be useful in ML. 1. Measure orthogonality of 2 vectors: Two vectors are said to be orthogonal of the angle between them is $90\degree$ and the outer product among them is maximum. @@ -106,35 +106,17 @@ A = \begin{bmatrix}a_1\\a_2\\a_3\\a_n\end{bmatrix} $$det(A-\lambda I)=0$$ whose roots give the eigen values of the Matrix $A$ as $\lambda _1$, $\lambda _2$, .. $\lambda _n$ 8. [M] Given the following matrix:
- $$ - \begin{bmatrix} - 1 & 4 & -2 \\ - -1 & 3 & 2 \\ - 3 & 5 & -6 - \end{bmatrix} - $$ +```math + \begin{bmatrix}1&4&-2\\-1&3&2\\3&5&-6\end{bmatrix} +``` Without explicitly using the equation for calculating determinants, what can we say about this matrix’s determinant? **Hint**: rely on a property of this matrix to determine its determinant. [A] The determinant of this matrix is $0$. We can perform some linear transformation to the columns and observe one of the columns can become all $0$s which will make the determinant to be $0$. - $$ - \begin{bmatrix} - 1 & 4 & -2 \\ - -1 & 3 & 2 \\ - 3 & 5 & -6 - \end{bmatrix} = -2\times - \begin{bmatrix} - 1 & 4 & 1 \\ - -1 & 3 & -1 \\ - 3 & 5 & 3 - \end{bmatrix} = \begin{bmatrix} - 1 & 4 & 0 \\ - -1 & 3 & 0 \\ - 3 & 5 & 0 - \end{bmatrix} - $$ - +```math + \begin{bmatrix}1&4&-2\\-1&3&2\\3&5&-6\end{bmatrix} = -2\times \begin{bmatrix}1&4&1\\-1&3&-1\\3&5&3\end{bmatrix} = \begin{bmatrix}1&4&0\\-1&3&0\\3&5&0\end{bmatrix} +``` 10. [M] What’s the difference between the covariance matrix $A^TA$ and the Gram matrix $AA^T$? 12. Given $A \in R^{n \times m}$ and $b \in R^n$ From dbae7272d78bbf07599e83501e1e9fc6a8a8525e Mon Sep 17 00:00:00 2001 From: JyotsnaT Date: Wed, 30 Aug 2023 22:04:46 +0530 Subject: [PATCH 12/21] 512 add space --- answers/chapter5.md | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/answers/chapter5.md b/answers/chapter5.md index 2b493b86..87802b3e 100644 --- a/answers/chapter5.md +++ b/answers/chapter5.md @@ -4,10 +4,11 @@ 1. [E] What’s the geometric interpretation of the dot product of two vectors?\ [A] In the geometric context, the dot product of 2 vecotrs is projection of one vector over another. For the given geometric vectors $\overrightarrow A$ and $\overrightarrow B$, their dot product is given by\ - $$\overrightarrow A.\overrightarrow B = |\overrightarrow A|.|\overrightarrow B|.cos\theta$$ + $$\overrightarrow A.\overrightarrow B = |\overrightarrow A|.|\overrightarrow B|.cos\theta$$ + ![](vectors1.png) - 2. [E] Given a vector $u$, find vector $v$ of unit length such that the dot product of $u$ and $v$ is maximum.\ + 3. [E] Given a vector $u$, find vector $v$ of unit length such that the dot product of $u$ and $v$ is maximum.\ [A] Dot product of $\vec{u}$ and $\vec{v}$ is givenby $|\vec{u}|.|\vec{v}|.cos\theta$. Where $|\vec{u}|$ and $\vec{v}$ are the magnitude of the vectors respectively. This dot product will be maximum when $cos\theta$ will be maximum, which occurs when $\theta = 0$ i.e both the vector overlap with each other.\ Since, $max |\vec{u}|.|\vec{v}|.cos\theta$\ $= |\vec{u}|.|\vec{v}|.1$\ From 6936675a8fb24eb2bd711d7e9b904a0edf34c6fc Mon Sep 17 00:00:00 2001 From: JyotsnaT Date: Wed, 30 Aug 2023 22:09:39 +0530 Subject: [PATCH 13/21] 512 add relative path --- answers/chapter5.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/answers/chapter5.md b/answers/chapter5.md index 87802b3e..591f539b 100644 --- a/answers/chapter5.md +++ b/answers/chapter5.md @@ -6,7 +6,7 @@ For the given geometric vectors $\overrightarrow A$ and $\overrightarrow B$, their dot product is given by\ $$\overrightarrow A.\overrightarrow B = |\overrightarrow A|.|\overrightarrow B|.cos\theta$$ - ![](vectors1.png) + ![vectors](./vectors1.png) 3. [E] Given a vector $u$, find vector $v$ of unit length such that the dot product of $u$ and $v$ is maximum.\ [A] Dot product of $\vec{u}$ and $\vec{v}$ is givenby $|\vec{u}|.|\vec{v}|.cos\theta$. Where $|\vec{u}|$ and $\vec{v}$ are the magnitude of the vectors respectively. This dot product will be maximum when $cos\theta$ will be maximum, which occurs when $\theta = 0$ i.e both the vector overlap with each other.\ From 2c73caf66ef26e5fb087556ddb53441eba4448e9 Mon Sep 17 00:00:00 2001 From: JyotsnaT Date: Wed, 30 Aug 2023 22:17:23 +0530 Subject: [PATCH 14/21] image folder added --- answers/chapter5.md | 2 +- answers/{ => imgs}/vectors1.png | Bin 2 files changed, 1 insertion(+), 1 deletion(-) rename answers/{ => imgs}/vectors1.png (100%) diff --git a/answers/chapter5.md b/answers/chapter5.md index 591f539b..813a173a 100644 --- a/answers/chapter5.md +++ b/answers/chapter5.md @@ -6,7 +6,7 @@ For the given geometric vectors $\overrightarrow A$ and $\overrightarrow B$, their dot product is given by\ $$\overrightarrow A.\overrightarrow B = |\overrightarrow A|.|\overrightarrow B|.cos\theta$$ - ![vectors](./vectors1.png) + ![vectors](imgs/vectors1.png) 3. [E] Given a vector $u$, find vector $v$ of unit length such that the dot product of $u$ and $v$ is maximum.\ [A] Dot product of $\vec{u}$ and $\vec{v}$ is givenby $|\vec{u}|.|\vec{v}|.cos\theta$. Where $|\vec{u}|$ and $\vec{v}$ are the magnitude of the vectors respectively. This dot product will be maximum when $cos\theta$ will be maximum, which occurs when $\theta = 0$ i.e both the vector overlap with each other.\ diff --git a/answers/vectors1.png b/answers/imgs/vectors1.png similarity index 100% rename from answers/vectors1.png rename to answers/imgs/vectors1.png From b3f9b3138de6a0e16bbbb6ba59a4c47a0430b015 Mon Sep 17 00:00:00 2001 From: JyotsnaT Date: Wed, 30 Aug 2023 22:25:16 +0530 Subject: [PATCH 15/21] change image import to html --- answers/chapter5.md | 9 ++++----- 1 file changed, 4 insertions(+), 5 deletions(-) diff --git a/answers/chapter5.md b/answers/chapter5.md index 813a173a..84265f7d 100644 --- a/answers/chapter5.md +++ b/answers/chapter5.md @@ -2,11 +2,10 @@ 1. Dot product 1. [E] What’s the geometric interpretation of the dot product of two vectors?\ - [A] In the geometric context, the dot product of 2 vecotrs is projection of one vector over another. - For the given geometric vectors $\overrightarrow A$ and $\overrightarrow B$, their dot product is given by\ + [A] In the geometric context, the dot product of 2 vecotrs is projection of one vector over another. For the given geometric vectors $\overrightarrow A$ and $\overrightarrow B$, their dot product is given by\ $$\overrightarrow A.\overrightarrow B = |\overrightarrow A|.|\overrightarrow B|.cos\theta$$ - - ![vectors](imgs/vectors1.png) +

+ 3. [E] Given a vector $u$, find vector $v$ of unit length such that the dot product of $u$ and $v$ is maximum.\ [A] Dot product of $\vec{u}$ and $\vec{v}$ is givenby $|\vec{u}|.|\vec{v}|.cos\theta$. Where $|\vec{u}|$ and $\vec{v}$ are the magnitude of the vectors respectively. This dot product will be maximum when $cos\theta$ will be maximum, which occurs when $\theta = 0$ i.e both the vector overlap with each other.\ @@ -17,7 +16,7 @@ $\vec{u}.\vec{v}$ = $= |\vec{u}|$\ $$\vec{v} = \frac{|\vec{u}|}{\vec{u}}$$ -2. Outer product +3. Outer product 1. [E] Given two vectors $a = [3, 2, 1]$ and $b = [-1, 0, 1]$. Calculate the outer product $a^Tb$?\ [A] Given \ $$a = [3, 2, 1], b = [-1, 0, 1]$$\ From ec0b7b7e0b2a8d226cc73eec7c2dd63686590503 Mon Sep 17 00:00:00 2001 From: JyotsnaT Date: Wed, 30 Aug 2023 22:32:48 +0530 Subject: [PATCH 16/21] 521 update --- answers/chapter5.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/answers/chapter5.md b/answers/chapter5.md index 84265f7d..ebd7741f 100644 --- a/answers/chapter5.md +++ b/answers/chapter5.md @@ -4,7 +4,7 @@ 1. [E] What’s the geometric interpretation of the dot product of two vectors?\ [A] In the geometric context, the dot product of 2 vecotrs is projection of one vector over another. For the given geometric vectors $\overrightarrow A$ and $\overrightarrow B$, their dot product is given by\ $$\overrightarrow A.\overrightarrow B = |\overrightarrow A|.|\overrightarrow B|.cos\theta$$ -

+

3. [E] Given a vector $u$, find vector $v$ of unit length such that the dot product of $u$ and $v$ is maximum.\ From 1e610c3bca2d479a56900cf21f2480e1253fdb84 Mon Sep 17 00:00:00 2001 From: JyotsnaT Date: Wed, 30 Aug 2023 22:45:44 +0530 Subject: [PATCH 17/21] rendering changes --- answers/chapter5.md | 15 ++++++++------- 1 file changed, 8 insertions(+), 7 deletions(-) diff --git a/answers/chapter5.md b/answers/chapter5.md index ebd7741f..63d97113 100644 --- a/answers/chapter5.md +++ b/answers/chapter5.md @@ -4,10 +4,11 @@ 1. [E] What’s the geometric interpretation of the dot product of two vectors?\ [A] In the geometric context, the dot product of 2 vecotrs is projection of one vector over another. For the given geometric vectors $\overrightarrow A$ and $\overrightarrow B$, their dot product is given by\ $$\overrightarrow A.\overrightarrow B = |\overrightarrow A|.|\overrightarrow B|.cos\theta$$ -

- - - 3. [E] Given a vector $u$, find vector $v$ of unit length such that the dot product of $u$ and $v$ is maximum.\ +

+ + +1. + 2. [E] Given a vector $u$, find vector $v$ of unit length such that the dot product of $u$ and $v$ is maximum.\ [A] Dot product of $\vec{u}$ and $\vec{v}$ is givenby $|\vec{u}|.|\vec{v}|.cos\theta$. Where $|\vec{u}|$ and $\vec{v}$ are the magnitude of the vectors respectively. This dot product will be maximum when $cos\theta$ will be maximum, which occurs when $\theta = 0$ i.e both the vector overlap with each other.\ Since, $max |\vec{u}|.|\vec{v}|.cos\theta$\ $= |\vec{u}|.|\vec{v}|.1$\ @@ -26,7 +27,7 @@ a^Tb = \begin{bmatrix}3\\2\\1\end{bmatrix} \times \begin{bmatrix}-1&0&1\end{bmatrix} = \begin{bmatrix}-3&0&3\\-2&0&2\\-1&0&1\end{bmatrix} ``` -a +2. 2. [M] Give an example of how the outer product can be useful in ML.\ [A] Following are the use cases where the outer product of the vectors can be useful in ML. 1. Measure orthogonality of 2 vectors: Two vectors are said to be orthogonal of the angle between them is $90\degree$ and the outer product among them is maximum. @@ -113,7 +114,7 @@ A = \begin{bmatrix}a_1\\a_2\\a_3\\a_n\end{bmatrix} Without explicitly using the equation for calculating determinants, what can we say about this matrix’s determinant? **Hint**: rely on a property of this matrix to determine its determinant. [A] The determinant of this matrix is $0$. - We can perform some linear transformation to the columns and observe one of the columns can become all $0$s which will make the determinant to be $0$. + We can perform some linear transformation to the columns and observe one of the columns can become all $0s$ which will make the determinant to be $0$. ```math \begin{bmatrix}1&4&-2\\-1&3&2\\3&5&-6\end{bmatrix} = -2\times \begin{bmatrix}1&4&1\\-1&3&-1\\3&5&3\end{bmatrix} = \begin{bmatrix}1&4&0\\-1&3&0\\3&5&0\end{bmatrix} ``` @@ -132,4 +133,4 @@ A = \begin{bmatrix}a_1\\a_2\\a_3\\a_n\end{bmatrix} 14. [H] Say we have the weights $w \in R^{d \times m}$ and a mini-batch $x$ of $n$ elements, each element is of the shape $1 \times d$ so that $x \in R^{n \times d}$. We have the output $y = f(x; w) = xw$. What’s the dimension of the Jacobian $\frac{\delta y}{\delta x}$? 15. [H] Given a very large symmetric matrix A that doesn’t fit in memory, say $A \in R^{1M \times 1M}$ and a function $f$ that can quickly compute $f(x) = Ax$ for $x \in R^{1M}$. Find the unit vector $x$ so that $x^TAx$ is minimal. - **Hint**: Can you frame it as an optimization problem and use gradient descent to find an approximate solution? \ No newline at end of file + **Hint**: Can you frame it as an optimization problem and use gradient descent to find an approximate solution? From 9c74cc0d319184b99a651f9a0597d5e2d784d00f Mon Sep 17 00:00:00 2001 From: JyotsnaT Date: Fri, 1 Sep 2023 12:33:44 +0530 Subject: [PATCH 18/21] 512 q 8 to 10 --- answers/chapter5.md | 14 ++++++++------ 1 file changed, 8 insertions(+), 6 deletions(-) diff --git a/answers/chapter5.md b/answers/chapter5.md index 63d97113..c537a5e8 100644 --- a/answers/chapter5.md +++ b/answers/chapter5.md @@ -93,11 +93,11 @@ A = \begin{bmatrix}a_1\\a_2\\a_3\\a_n\end{bmatrix} [A] The determinant of the matrix is given by $$det(A) = |A| = \sum_{i=1}^{n}A_{ij}C_{ij}$$ where $C_{ij}$ is the cofactor matrix. Determinant is only defined for square matrices. It represents the area/volume enclosed between the vectors. -5. [E] What happens to the determinant of a matrix if we multiply one of its rows by a scalar $t \times R$?\ +4. [E] What happens to the determinant of a matrix if we multiply one of its rows by a scalar $t \times R$?\ [A] As per the row scaling property, on multiplying one of the rows with a scalar $t$ the resulting determinant will be scaled by $t$. $$det(t\times A) = t\times det(A)$$ -6. [M] A $4 \times 4$ matrix has four eigenvalues $3, 3, 2, -1$. What can we say about the trace and the determinant of this matrix?\ +5. [M] A $4 \times 4$ matrix has four eigenvalues $3, 3, 2, -1$. What can we say about the trace and the determinant of this matrix?\ [A] Trace of a matrix is given by the sum of it's eigen values.\ Trace of the matrix : For a diagonal matrix, the sum of all it's diagonal elements. $$Tr(A) = \sum_{i=1}^{n}a_{ij}$$ @@ -106,7 +106,7 @@ A = \begin{bmatrix}a_1\\a_2\\a_3\\a_n\end{bmatrix} $\lambda$ is the eigen value of $A$ and $X$ is eigen vector of A given $X != 0$. The charecteristic equation of $A$ is given by $$det(A-\lambda I)=0$$ whose roots give the eigen values of the Matrix $A$ as $\lambda _1$, $\lambda _2$, .. $\lambda _n$ -8. [M] Given the following matrix:
+6. [M] Given the following matrix:
```math \begin{bmatrix}1&4&-2\\-1&3&2\\3&5&-6\end{bmatrix} ``` @@ -118,10 +118,12 @@ A = \begin{bmatrix}a_1\\a_2\\a_3\\a_n\end{bmatrix} ```math \begin{bmatrix}1&4&-2\\-1&3&2\\3&5&-6\end{bmatrix} = -2\times \begin{bmatrix}1&4&1\\-1&3&-1\\3&5&3\end{bmatrix} = \begin{bmatrix}1&4&0\\-1&3&0\\3&5&0\end{bmatrix} ``` -10. [M] What’s the difference between the covariance matrix $A^TA$ and the Gram matrix $AA^T$? +7. [M] What’s the difference between the covariance matrix $A^TA$ and the Gram matrix $AA^T$?\ + [A] The covariance matrix $A^TA$ is the pair wise inner product of the features/dimensions of the vector space. The diagonal of this matrix is variance of each feature. It is helpful in undertanding the relation of features with each other and can be applied to areas like feature transformation, feature selection, dimensionality reduction, etc. The Gram matrix $AA^T$ is the pair wise inner product of each vector. It is essential in understanding the relation between individual vectors. This is primarily applied in Kernel methods as a kernel function between data points and other applications where similarity between data points in high dimensional space needs to be calculated. -12. Given $A \in R^{n \times m}$ and $b \in R^n$ - 1. [M] Find $x$ such that: $Ax = b$. +8. Given $A \in R^{n \times m}$ and $b \in R^n$ + 1. [M] Find $x$ such that: $Ax = b$.\ + [A] 1. [E] When does this have a unique solution? 1. [M] Why is it when A has more columns than rows, $Ax = b$ has multiple solutions? 1. [M] Given a matrix A with no inverse. How would you solve the equation $Ax = b$? What is the pseudoinverse and how to calculate it? From 901c9511b9847a672c89394031552ab8cd6ccd60 Mon Sep 17 00:00:00 2001 From: JyotsnaT Date: Fri, 1 Sep 2023 20:31:41 +0530 Subject: [PATCH 19/21] 512 added 8 and 9 solution --- answers/chapter5.md | 49 ++++++++++++++++++++++++++++++++++++++++----- 1 file changed, 44 insertions(+), 5 deletions(-) diff --git a/answers/chapter5.md b/answers/chapter5.md index c537a5e8..9203d9ca 100644 --- a/answers/chapter5.md +++ b/answers/chapter5.md @@ -123,14 +123,53 @@ A = \begin{bmatrix}a_1\\a_2\\a_3\\a_n\end{bmatrix} 8. Given $A \in R^{n \times m}$ and $b \in R^n$ 1. [M] Find $x$ such that: $Ax = b$.\ - [A] - 1. [E] When does this have a unique solution? - 1. [M] Why is it when A has more columns than rows, $Ax = b$ has multiple solutions? - 1. [M] Given a matrix A with no inverse. How would you solve the equation $Ax = b$? What is the pseudoinverse and how to calculate it? - + [A] The solution to x can be given the following when $A$ is a square matrix, i.e, it't determinant exists. + $$x = A^{-1}b$$ + 1. [E] When does this have a unique solution?\ + [A] The solution for this equation will not have a unique solution when the matrix $A$ is not invertible. + 1. [M] Why is it when A has more columns than rows, $Ax = b$ has multiple solutions?\ + [A] When $A$ has more columns than rows, the matrrix $A$ becomes singular matrix and it's determinant is undefined. In that case, either the solution or above equation is non existant or there are infinitely many solutions. + 1. [M] Given a matrix A with no inverse. How would you solve the equation $Ax = b$? What is the pseudoinverse and how to calculate it?\ + [A] The solution for equation $Ax = b$ can always be approximated. One such approximation is defined by replacing the actual inverse with a pseudoinverse. + $$x = A^+b$$ + The pseudo inverse also known as 'Moore-penrose inverse' can be calculated through SVD(Singular valued decomposition) of matrix $A$ which decomposed the matrix $A$ as + $$A = U\Sigma V^T$$ + Where $U$ is a matrix of dimension $m\times m$, $\Sigma$ is a diagonal matrix of dimension $m\times n$ where the diagonal values are called as the singular values and $V$ is a matrix of dimension $n\times n$. The pseudo inverse of $A$ can be written as + $$A^+ = V\Sigma^+ U^T$$ + $\Sigma^+$ can be found by reciprocrating the non-zero singular values in $\Sigma$ followed by transposition. + 13. Derivative is the backbone of gradient descent. 1. [E] What does derivative represent? + [A] In case of gradient descent, we compute gradient of the cost function which is partial derivative of the cost function w.r.t each parameter. The derivative represents the direction of the vector denoting highest increase in cost value also known as the 'slope of the curve' or 'rate of change of cost function'. 1. [M] What’s the difference between derivative, gradient, and Jacobian? + [A] These 3 are all inter-related terms in linear algebra + Derivative or first order derivative or differential is a scalar and defined for a univariate function $f(x)$ and is represented as + $$f'(x) = \frac{df}{dx} = \lim\limits_{h\to \infty}\frac{f(x+h) - f(x)}{h}$$ + Gradient is a generalization term for derivative and is defined for multivariate functions. It is given by a vector of partial differential of the function w.r.t. each variable. For a function $f(x_1, x_2)$ the gradient is given by + $$\nabla_x f = [\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}]$$ + Jacobian is a matrix of partial differentials for a vector valued function $f: R^n \rightarrow R^m$. For a vector valued function +```math +f(x) = +\begin{bmatrix} +f_1(x) \\ +f_2(x) \\ +\vdots \\ +f_m(x) +\end{bmatrix} +``` + +the jacobian $J$ will be given by + +```math +J = \nabla_x f = \begin{bmatrix} \frac{\partial f(x)}{\partial x_1} & \dots & \frac{\partial f(x)}{\partial x_n} \end{bmatrix} + += \begin{bmatrix} +\frac{\partial f_1(x)}{\partial x_1} & \dots & \frac{\partial f_1(x)}{\partial x_n} \\ +\frac{\partial f_2(x)}{\partial x_1} & \dots & \frac{\partial f_2(x)}{\partial x_n} \\ +\vdots & & \vdots\\ +\frac{\partial f_m(x)}{\partial x_1} & \dots & \frac{\partial f_m(x)}{\partial x_n} +\end{bmatrix} +``` 14. [H] Say we have the weights $w \in R^{d \times m}$ and a mini-batch $x$ of $n$ elements, each element is of the shape $1 \times d$ so that $x \in R^{n \times d}$. We have the output $y = f(x; w) = xw$. What’s the dimension of the Jacobian $\frac{\delta y}{\delta x}$? 15. [H] Given a very large symmetric matrix A that doesn’t fit in memory, say $A \in R^{1M \times 1M}$ and a function $f$ that can quickly compute $f(x) = Ax$ for $x \in R^{1M}$. Find the unit vector $x$ so that $x^TAx$ is minimal. From b4cf067ebd34220288a62423928039d7ca89d558 Mon Sep 17 00:00:00 2001 From: JyotsnaT Date: Sat, 23 Sep 2023 19:49:57 +0530 Subject: [PATCH 20/21] Finish answers for matrices --- answers/chapter5.md | 69 ++++++++++++++++++++++++++++++--------------- 1 file changed, 46 insertions(+), 23 deletions(-) diff --git a/answers/chapter5.md b/answers/chapter5.md index 9203d9ca..e1a0236a 100644 --- a/answers/chapter5.md +++ b/answers/chapter5.md @@ -26,7 +26,7 @@ ```math a^Tb = \begin{bmatrix}3\\2\\1\end{bmatrix} \times \begin{bmatrix}-1&0&1\end{bmatrix} = \begin{bmatrix}-3&0&3\\-2&0&2\\-1&0&1\end{bmatrix} ``` - + 2. 2. [M] Give an example of how the outer product can be useful in ML.\ [A] Following are the use cases where the outer product of the vectors can be useful in ML. @@ -54,18 +54,31 @@ A = \begin{bmatrix}a_1\\a_2\\a_3\\a_n\end{bmatrix} [A] Dimension of the span of $n$ vectors is given by the rank of their augmented matrix. -9. Norms and metrics - 1. [E] What's a norm? What is $L_0, L_1, L_2, L_{norm}$?\ - [A] - 1. $L_0$ norm: Number of non zero elements in a vector. - 2. $L_1$ norm: Sum of absolute value of vector elements.\ - $|x|_1 = \sum{|x_i|}$ - 4. $L_2$ norm: Length of vector in Euclidean space.\ - $||x||_2 = (\sum{x_i^2})^{\frac{1}{2}}$ - 5. $L_{\infty}$ norm: Maximum absolute value of vector elemets.\ - $||x||_{\infty} = max{|x_i|}$ - 1. [M] How do norm and metric differ? Given a norm, make a metric. Given a metric, can we make a norm?\ - [A] +8. Norms and metrics + 1. [E] What's a norm? What is $L_0, L_1, L_2, L_{norm}$?\ + [A] For a given vector space, $V$, the norm is a function $n: V \rightarrow R$ defined to measure the length of a vector. Norm has following properties - + 1. Non Zero norm : $n(x) \ge 0$ for all $x \in V$ and $x \neq 0$ + 2. Scale invariant : $n(\lambda \times x) = |\lambda| \times n(x)$ for all $x \in V$ and for all $\lambda \in R$. + 3. Triangle inequality : $n(x + y) \le n(x) + n(y)$ for all $x,y \in V$ + + Norm is represented by $||x|| = n(x)$. These are the different types of norms + + 1. $L_0$ norm: Number of non zero elements in a vector. + 2. $L_1$ norm (Manhattan Norm): Sum of absolute value of vector elements.\ + $||x||_1 = \sum{|x_i|}$ + 3. $L_2$ norm (Eucledian Norm): Length of vector in Euclidean space.\ + $||x||_2 = (\sum{x_i^2})^{\frac{1}{2}}$ + 4. $L_{\infty}$ norm (Maximum norm): Maximum absolute value of vector elemets.\ + $||x||_{\infty} = max{|x_i|}$ + 2. [M] How do norm and metric differ? Given a norm, make a metric. Given a metric, can we make a norm?\ + [A] For a given vector space $V$, metrics is the function $m: V \times V \rightarrow R$ defined to measure the distance between two points in that vector space. The metrics has following properties - + 1. Non Zero Metric : $m(x,y) \ge 0 \forall x,y \in V$\ + $m(x,y) = 0$ when $x = y$ + 2. Symmetric property : $m(x,y) = m(y,x) \forall x,y \in V$ + 3. Triangle inequality : $m(x,y) \le m(x,z) + m(z,y) \forall x,y,z \in V$ + + A metric can be computed using a norm. Given two points $x, y$. The norm of difference of two vectors gives the metric of these points. + $$||x-y|| = m(x,y)$$ @@ -110,10 +123,10 @@ A = \begin{bmatrix}a_1\\a_2\\a_3\\a_n\end{bmatrix} ```math \begin{bmatrix}1&4&-2\\-1&3&2\\3&5&-6\end{bmatrix} ``` - - Without explicitly using the equation for calculating determinants, what can we say about this matrix’s determinant? - **Hint**: rely on a property of this matrix to determine its determinant. - [A] The determinant of this matrix is $0$. + + Without explicitly using the equation for calculating determinants, what can we say about this matrix’s determinant? + **Hint**: rely on a property of this matrix to determine its determinant.\ + [A] The determinant of this matrix is $0$. We can perform some linear transformation to the columns and observe one of the columns can become all $0s$ which will make the determinant to be $0$. ```math \begin{bmatrix}1&4&-2\\-1&3&2\\3&5&-6\end{bmatrix} = -2\times \begin{bmatrix}1&4&1\\-1&3&-1\\3&5&3\end{bmatrix} = \begin{bmatrix}1&4&0\\-1&3&0\\3&5&0\end{bmatrix} @@ -138,10 +151,10 @@ A = \begin{bmatrix}a_1\\a_2\\a_3\\a_n\end{bmatrix} $$A^+ = V\Sigma^+ U^T$$ $\Sigma^+$ can be found by reciprocrating the non-zero singular values in $\Sigma$ followed by transposition. -13. Derivative is the backbone of gradient descent. - 1. [E] What does derivative represent? +9. Derivative is the backbone of gradient descent. + 1. [E] What does derivative represent?\ [A] In case of gradient descent, we compute gradient of the cost function which is partial derivative of the cost function w.r.t each parameter. The derivative represents the direction of the vector denoting highest increase in cost value also known as the 'slope of the curve' or 'rate of change of cost function'. - 1. [M] What’s the difference between derivative, gradient, and Jacobian? + 1. [M] What’s the difference between derivative, gradient, and Jacobian?\ [A] These 3 are all inter-related terms in linear algebra Derivative or first order derivative or differential is a scalar and defined for a univariate function $f(x)$ and is represented as $$f'(x) = \frac{df}{dx} = \lim\limits_{h\to \infty}\frac{f(x+h) - f(x)}{h}$$ @@ -171,7 +184,17 @@ J = \nabla_x f = \begin{bmatrix} \frac{\partial f(x)}{\partial x_1} & \dots & \f \end{bmatrix} ``` -14. [H] Say we have the weights $w \in R^{d \times m}$ and a mini-batch $x$ of $n$ elements, each element is of the shape $1 \times d$ so that $x \in R^{n \times d}$. We have the output $y = f(x; w) = xw$. What’s the dimension of the Jacobian $\frac{\delta y}{\delta x}$? -15. [H] Given a very large symmetric matrix A that doesn’t fit in memory, say $A \in R^{1M \times 1M}$ and a function $f$ that can quickly compute $f(x) = Ax$ for $x \in R^{1M}$. Find the unit vector $x$ so that $x^TAx$ is minimal. +14. [H] Say we have the weights $w \in R^{d \times m}$ and a mini-batch $x$ of $n$ elements, each element is of the shape $1 \times d$ so that $x \in R^{n \times d}$. We have the output $y = f(x; w) = xw$. What’s the dimension of the Jacobian $\frac{\delta y}{\delta x}$?\ + [A] Since, $y$ is a vector valued function $f(x; w) \in R^{n \times m}$, the Jacobian will be a matrix of dimension $m \times n$ as described in the solution above. +16. [H] Given a very large symmetric matrix A that doesn’t fit in memory, say $A \in R^{1M \times 1M}$ and a function $f$ that can quickly compute $f(x) = Ax$ for $x \in R^{1M}$. Find the unit vector $x$ so that $x^TAx$ is minimal. - **Hint**: Can you frame it as an optimization problem and use gradient descent to find an approximate solution? + **Hint**: Can you frame it as an optimization problem and use gradient descent to find an approximate solution?\ + [A] This can be posed as a constrained optimization problem\ + $$\min x^{T}Ax$$ + $$||x|| = 1$$ + In an unconstrained optimization problem, gradient descent can be applied by updating the optimal $x$ in every step with gradient of the function. + $$x_{k+1} = x_k - \eta \nabla f(x^TAx)$$ + + In this case, there is constraint to the optimal value $x^*$. To solve this we can add an additional step to the gradient descent algorithm which projects the optimal value at every step to the constaint space $C$. This projection function, searches for the point on the unit sphere which is closes to the optimal point in that step. + $$x_{k+1} = P(x_k - \eta \nabla f(x^TAx))$$ + $$P(x) = \arg\min_{z \in C} |x - z|$$ From 60be1430de7015f6888c1f14eb772ee34cf780a8 Mon Sep 17 00:00:00 2001 From: Jyotsna Thapliyal Date: Wed, 12 Jun 2024 15:24:49 +0300 Subject: [PATCH 21/21] changed lim inf to lim 0 --- answers/chapter5.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/answers/chapter5.md b/answers/chapter5.md index e1a0236a..e8bd41c4 100644 --- a/answers/chapter5.md +++ b/answers/chapter5.md @@ -157,7 +157,7 @@ A = \begin{bmatrix}a_1\\a_2\\a_3\\a_n\end{bmatrix} 1. [M] What’s the difference between derivative, gradient, and Jacobian?\ [A] These 3 are all inter-related terms in linear algebra Derivative or first order derivative or differential is a scalar and defined for a univariate function $f(x)$ and is represented as - $$f'(x) = \frac{df}{dx} = \lim\limits_{h\to \infty}\frac{f(x+h) - f(x)}{h}$$ + $$f'(x) = \frac{df}{dx} = \lim\limits_{h\to 0}\frac{f(x+h) - f(x)}{h}$$ Gradient is a generalization term for derivative and is defined for multivariate functions. It is given by a vector of partial differential of the function w.r.t. each variable. For a function $f(x_1, x_2)$ the gradient is given by $$\nabla_x f = [\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}]$$ Jacobian is a matrix of partial differentials for a vector valued function $f: R^n \rightarrow R^m$. For a vector valued function