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Linear algebra is a fundamental pillar of data science. In advanced models in data science, like neural networks, the inputs and transformations are based upon vectors, matrices, and tensors which require a reasonable understanding of linear algebra to get the desired results. It is elegant and the most applied mathematics under the umbrella of data science.

This course teaches linear algebra with a focus on data science. This course encompasses several engaging illustrations, including static images and animations. Furthermore, this course presents mathematical modeling through programming in Python. This course contains several executable coding playgrounds on real data sets and a final project with practical applications.

Aside from theoretical implementations, the modern-day world needs its daunting calculations, trajectory mapping, and distance manipulation, all of which linear algebra provides. By the end of this course, you’ll have a working knowledge of all the necessary teachings in linear algebra.

Linear Algebra for Data Science Using Python

$
\left(\begin{array}{ccc|c}  
1 & -3 & 5 & 0\\ 
1 & 2 & -1 & 9\\ 
1 & -6 & 1 &-11
\end{array}\right)
$


$
\left(\begin{array}{ccc|c}  
1 & -3 & 5 & 0\\ 
0 & 2 & -1 & 9\\ 
1 & -6 & 0 &-11
\end{array}\right)
$


$
\left(\begin{array}{ccc|c}  
1 & 3 & 5 & 0\\ 
0 & 2 & 1 & 9\\ 
1 & 6 & 0 &11
\end{array}\right)
$


$
\left(\begin{array}{ccc|c}  
1 & -3 & 5 & 0\\ 
0 & 2 & -1 & 9\\ 
1 & -6 & 0 &11
\end{array}\right)
$

Which is the correct augmented matrix for the following system?

$x_1-3x_2+5x_3=0$

$2x_2-x_3=9$

$x_1-6x_2=-11$

Which is the correct augmented matrix for the following system?
<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>−</mo><mn>3</mn><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mn>5</mn><msub><mi>x</mi><mn>3</mn></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x_1-3x_2+5x_3=0</annotation></semantics></math>x1​−3x2​+5x3​=0
<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>2</mn><msub><mi>x</mi><mn>2</mn></msub><mo>−</mo><msub><mi>x</mi><mn>3</mn></msub><mo>=</mo><mn>9</mn></mrow><annotation encoding="application/x-tex">2x_2-x_3=9</annotation></semantics></math>2x2​−x3​=9
<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>−</mo><mn>6</mn><msub><mi>x</mi><mn>2</mn></msub><mo>=</mo><mo>−</mo><mn>11</mn></mrow><annotation encoding="application/x-tex">x_1-6x_2=-11</annotation></semantics></math>x1​−6x2​=−11

$
\begin{bmatrix}  
1 & 3 & 0 & 0\\ 
0 & 2 & -1 & 3\\ 
0 & 0 & 0 & 21
\end{bmatrix}
$

$
\begin{bmatrix}  
0 & 1 & 2 & 0 & 3\\ 
0 & 0 & 0 & 1 & 0 \\ 
0 & 0 & 0 & 0 & 0
\end{bmatrix}
$

$
\begin{bmatrix}  
0 & 1 & 0 & 0 \\ 
0 & 1 & 0 & 1 \\ 
0 & 0 & 0 & 1 
\end{bmatrix}
$

$
\begin{bmatrix}  
0 & 0 & 0 & 1\\ 
0 & 0 & 0 & 0
\end{bmatrix}
$

(Select all that apply.) Which matrices are in row echelon form? 

(Select all that apply.) Which matrices are in row echelon form?

Number of columns minus the total number of rows $(n-m)$

Number of columns minus the number of free variables in $rref(A)$

(Select all that apply.) Which statements correctly define the rank of a matrix  $A$? (Choose all that apply)

(Select all that apply.) Which statements correctly define the rank of a matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>A? (Choose all that apply)

What’s the largest possible rank of a $(3\times5)$ matrix?

What’s the largest possible rank of a <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mn>3</mn><mo>×</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(3\times5)</annotation></semantics></math>(3×5) matrix?

(Fill in the blank.) A system of linear equations with free variables in its  $rref$ has ______.

(Fill in the blank.) A system of linear equations with free variables in its <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi><mi>r</mi><mi>e</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">rref</annotation></semantics></math>rref has ______.

Select values of $a$, $b$, $c$, and $d$, such that the augmented matrix
$$[A|b] = \left(\begin{array}{ccc|c}
1 & 2 & 3 & a\\
0 & 4 & 5 & b\\
0 & 0 & d & c 
\end{array}\right)$$
has no solution.

Select values of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math>a, <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math>b, <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>c, and <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>d, such that the augmented matrix
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">[</mo><mi>A</mi><mi mathvariant="normal">∣</mi><mi>b</mi><mo stretchy="false">]</mo><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center center center" columnlines="none none solid" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>3</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>4</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>5</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">[A|b] = \left(\begin{array}{ccc|c}
1 &amp; 2 &amp; 3 &amp; a\\
0 &amp; 4 &amp; 5 &amp; b\\
0 &amp; 0 &amp; d &amp; c 
\end{array}\right)</annotation></semantics></math>[A∣b]=⎝<svg xmlns="http://www.w3.org/2000/svg" width='0.875em' height='0.016em' style='width:0.875em' viewBox='0 0 875 16' preserveAspectRatio='xMinYMin'><path d='M291 0 H417 V16 H291z M291 0 H417 V16 H291z'/></svg>⎛​100​240​35d​abc​⎠<svg xmlns="http://www.w3.org/2000/svg" width='0.875em' height='0.016em' style='width:0.875em' viewBox='0 0 875 16' preserveAspectRatio='xMinYMin'><path d='M457 0 H583 V16 H457z M457 0 H583 V16 H457z'/></svg>⎞​
has no solution.

Select values of $a$, $b$, $c$, and $d$, such that the augmented matrix
$$[A|b] = \left(\begin{array}{ccc|c}
1 & 2 & 3 & a\\
0 & 4 & 5 & b\\
0 & 0 & d & c 
\end{array}\right)$$
has infinite solutions.

Select values of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math>a, <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math>b, <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>c, and <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>d, such that the augmented matrix
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">[</mo><mi>A</mi><mi mathvariant="normal">∣</mi><mi>b</mi><mo stretchy="false">]</mo><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center center center" columnlines="none none solid" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>2</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>3</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>a</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>4</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>5</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>b</mi></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>d</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>c</mi></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">[A|b] = \left(\begin{array}{ccc|c}
1 &amp; 2 &amp; 3 &amp; a\\
0 &amp; 4 &amp; 5 &amp; b\\
0 &amp; 0 &amp; d &amp; c 
\end{array}\right)</annotation></semantics></math>[A∣b]=⎝<svg xmlns="http://www.w3.org/2000/svg" width='0.875em' height='0.016em' style='width:0.875em' viewBox='0 0 875 16' preserveAspectRatio='xMinYMin'><path d='M291 0 H417 V16 H291z M291 0 H417 V16 H291z'/></svg>⎛​100​240​35d​abc​⎠<svg xmlns="http://www.w3.org/2000/svg" width='0.875em' height='0.016em' style='width:0.875em' viewBox='0 0 875 16' preserveAspectRatio='xMinYMin'><path d='M457 0 H583 V16 H457z M457 0 H583 V16 H457z'/></svg>⎞​
has infinite solutions.

For the following vector equation
$$
x_1 \begin{bmatrix} 1 \\ 3 \\ 5 \end{bmatrix} +
x_2 \begin{bmatrix} 2 \\ 5 \\ 9 \end{bmatrix} +
x_3 \begin{bmatrix} -3 \\ 9 \\ 3 \end{bmatrix}
= \begin{bmatrix} 0 \\ 0  \\ 0  \end{bmatrix}
$$
and the $rref$
$$\left( \begin{array}{ccc|c}
1 & 0 & 33 & 0 \\
0 & 1 & -18 & 0 \\
0 & 0 &  0 & 0 \\
\end{array}\right),$$ 
which of the following is a valid combination of scalars $(x_1, x_2, x_3)$?

Introduction

Linearity

Matrices

Solving Linear Systems

Singularity

Linear Regression and Least Squares

Vector Space

Vector Spaces of a Matrix

Singular Value Decomposition: SVD

Learning to Find Discriminative Null Space for Face Recognition

Quiz: Linear Systems