studybyyourself » Chapter 1 : Introduction

1. Goal

The goal of linear algebra is to solve systems of linear equations, and tell if the solution is unique or not. Besides, linear algebra aims to transform spaces.

Objects

In elementary algebra operations are performed on numbers, e.g.:

$\begin{equation} 2x = 10 \iff x = 5 \end{equation}$

As we’ll see, in linear algebra operations are performed on matrices and vectors, e.g. :

$\begin{equation} \begin{pmatrix} 1 & 4 \\ 0 & 2 \end{pmatrix}\,\vec{x} = \begin{pmatrix} 1 \\ 1 \end{pmatrix} \iff \vec{x} = \begin{pmatrix} -1 \\ \frac{1}{2} \end{pmatrix} \end{equation}$

Recapitulation

In linear algebra operations are no longer performed on numbers but on matrices and vectors. These one are a generalization of the concept of number.

Indeed, in $(2)$, the vector $\vec{x}$ has two entries. We can see numbers as vectors but having one single element.