Crux of linear algebra

Question

What is the connection between solving systems of linear equations and vector spaces? And what do matrices have to do with all of that?

I know this is a crux of linear algebra, and therefore, not so specific question but I need a motivation approach for what linear algebra is. Thanks.

Answer 1

A system of linear equations is a vector equation that expresses that you know the image of some vector by a given linear transformation, and you are looking for the components that unknown vector.

Consider a linear transformation from $\mathbb R^n$ to $\mathbb R^m$. Any vector of $\mathbb R^n$ can be expressed as a linear combination of the elements of the canonical basis (unit vectors along every axis). And by linearity of the transformation, the image of a linear combination is the same linear combination applied to the images. (This is where vector spaces appear.)

Now the images of the canonical vectors are $n$ vectors of dimension $m$, and the above reasoning shows that the linear transformation is completely determined by $n\times m$ numbers. These numbers can be arranged in a 2D array. (This is where matrices appear.)

Answer 2

Vector spaces and linear maps arise in a lot of ways when dealing with systems of linear equations:

Note that you can write any linear system as $Ax = b$ where $A \in \Bbb R^{n,m}, x \in \Bbb R^m, b \in \Bbb R^n$ (considering the simple case where we have a linear system with real numbers). Assume you have some solution $x_0$ of this system and another vector $\hat{x}$ such that $A \hat{x} = 0$, then clearly $x_0+\alpha\hat{x}$ is also a solution of your system since $A(x_0 + \alpha\hat{x}) = Ax_0 + \alpha A\hat{x} = b + \alpha 0 = b$ for all $\alpha \in \Bbb R$.

Note that in particular for the case $b=0$ we have that you can multiply any two solutions by some factor, add them up and you'll still have a solution for the original system. So the set of all solutions of $Ax=0$ is a vector space - and the set of all solutions of $Ax=b$ is a so called affine space where the underlying vector space is precisely the solution set of $Ax=0$.