1. Matrix and Linear Systems

2026-01-21

What is Linear Algebra?

To put it simple, Linear Algebra is the study of linear transformations.

Transformation transforms an input to an output. In other words, a transformation is a function.

e.g., $f(x) = 2x$ is a transformation that transforms input $x$ to output $2x$ .

A transformation $T$ that satisfies the linearity property:

Let

[m] \times [n] = \set{ (i, j) | 1 \leq i \leq m, 1 \leq j \leq n, i, j \in Z }

then an $m \times n$ (real) matrix is a function

A: [m] \times [n] \to R

We write the output by $A(i, j) = a_{i, j}$ , where

In general, a matrix is represented as

A = \begin{bmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m,1} & a_{m,2} & \cdots & a_{m,n} \end{bmatrix}

A vector is an $m \times 1$ matrix (a special matrix that has only one column):

\vec{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_m \end{bmatrix}