Linear Systems - Introduction

Recall that we introduced the idea of the matrix-by-vector product as being a linear combination of the columns of a matrix and the components of a vector.

Take the vectors

Their linear combinations are denoted by $c \vec{u} + d \vec{v} + e \vec{w}$, that is

We now rewrite this linear combination as a matrix-by-vector product. We put the vectors in the columns of the matrix $A$ and put the scalars $c$, $d$, and $e$ in the column vector $\vec{x}$. That is

Performing the matrix–vector multiplication produces exactly this linear combination of the columns of $A$: $c \vec{u} + d \vec{v} + e \vec{w}$.

Conventionally, the resulting vector is denoted by $\vec{b}$.

Up until now, when performing matrix-by-vector multiplication, we were given a matrix $A$ and a vector $\vec{x}$ to multiply with, and we were tasked with finding the resulting vector $\vec{b}$. Now we change our goal: we are given some matrix $A$ and the vector $\vec{b}$ which is the result of the product $A \vec{x}$, and we are tasked with finding the vector $\vec{x}$ which leads to that result. We are basically “reversing” the problem.

So before we were asking the question “what is the result of the linear combination $c \vec{u} + d \vec{v} + e \vec{w}$?”, and now we are asking the question “what linear combination of $\vec{u}$, $\vec{v}$, and $\vec{w}$ results in the result $\vec{b}$?”.

The expression $A\vec{x} = \vec{b}$ is referred to as a linear system, and solving the linear system means finding the vector $\vec{x}$ such that $A\vec{x} = \vec{b}$.

Conventionally, instead of $c$, $d$ and $e$, the symbols $x_1$, $x_2$ and $x_3$ are used to denote the unknowns. This is also coherent with the notation used for denoting the components of a vector.

Similarly, the components of the resulting vector $\vec{b}$ are referred to as $b_1$, $b_2$, $b_3$, and so on.

In the context of linear systems, the columns of a matrix $A$ are often denoted by $\vec{a}_1, \vec{a}_2, \cdots$, as opposed to the usual notation of $A_{\star 1}, A_{\star 2}, \cdots$. This new notation really emphasizes that the columns of a matrix are simply column vectors.

Example

Take the linear system

Solving the system would mean finding the vector

This means that the vector $\vec{b}$ can be rewritten as

We can even check the solution: