Linear Algebra: Deep Learning Book

Linear algebra is a form of continuous rather than discrete mathematics.

Scalars, Vectors, Matrices and Tensors

$I_n \in \mathbb{R}^{n \times n}$ ; where main diagonal = 1's and all other entries = 0's
An identity matrix $I_n$ is a matrix that does not change any vector $x$ when we multiply that vector by that matrix. $I_nx = x$

Ax=b \newline Step \_1: Multiply \ by \ A^{-1} \newline A^{-1}Ax=A^{-1}b \newline Step \_2: A^{-1}A = I_n \newline I_nx = A^{-1}b \newline Step \_3: Multiplying\ with \ I_n \ will \ not \ change \ the \ vector \newline x = A^{-1}b

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Identity and Inverse Matrix

Identity Matrix

$I_n \in \mathbb{R}^{n \times n}$ ; where main diagonal = 1's and all other entries = 0's

An identity matrix $I_n$ is a matrix that does not change any vector $x$ when we multiply that vector by that matrix. $I_nx = x$

Inverse Matrix

Denoted by $A^{-1}$ ; where $A^{-1}A = I_n$

Use case: $A^{-1}$ can be used to solve linear equations:

Ax=b \newline Step \_1: Multiply \ by \ A^{-1} \newline A^{-1}Ax=A^{-1}b \newline Step \_2: A^{-1}A = I_n \newline I_nx = A^{-1}b \newline Step \_3: Multiplying\ with \ I_n \ will \ not \ change \ the \ vector \newline x = A^{-1}b