studybyyourself » Chapter 1 : Introduction

1. Goal

The primary goal of linear algebra is to find the solutions of systems of linear equations. Such a system may have a unique solution (a determined system), infinitely many solutions (an indeterminate system), or no solution (an inconsistent system).

Linear algebra also deals with geometric transformations such as reflections, central symmetry, rotations, scalings, shears, and orthogonal projections.

2. 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. :

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

Wrap-up

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

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