Linear combination of columns

Consider the system given by $Ax=b$ where $A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]$, $x=\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]$, and $b=\left[\begin{array}{c}7\\ 8\end{array}\right]$.

Recall that $Ax$ in this case denotes the tuple $\left[\begin{array}{c}{x}_1+2x_2+3x_3\\ 4x_1+5x_2+6x_3\end{array}\right]$.

Using tuple arithmetic, this tuple can be written as $x_1\left[\begin{array}{c}1\\ 4\end{array}\right]+x_2\left[\begin{array}{c}2\\ 5\end{array}\right]+x_3\left[\begin{array}{c}3\\ 6\end{array}\right]$.

What we have here is called a linear combination of the tuples $\left[\begin{array}{c}1\\ 4\end{array}\right]$, $\left[\begin{array}{c}2\\ 5\end{array}\right]$, and $\left[\begin{array}{c}3\\ 6\end{array}\right]$. (In general, a linear combination of the tuples $t_1,\dots ,t_k$ has the form $a_1{t}_1+a_2{t}_2+\cdots +a_k{t}_k$ where $a_i$ is a scalar for each $i=1,\dots ,k$.)

The question of whether or not $Ax=b$ has a solution can be interpreted as follows: Is $b$ a linear combination of the columns of $A$?

In general, if $A=[A_1\cdots A_n]$ where $A_i$ is the $i$th column of $A$ and $x=\left[\begin{array}{c}{x}_1\\ ⋮\\ x_n\end{array}\right]$, then $Ax=x_1{A}_1+x_2{A}_2+\cdots +x_n{A}_n$. Solving $Ax=b$ means finding a linear combination of $A_1,\dots ,A_n$ that gives $b$.

Quick Quiz

Exercises

For each of the following, write it as a linear combination of tuples with $x,y,z$ as scalars.

1. $\left[\begin{array}{c}x+2y-z\\ y+z\\ 2x\end{array}\right]$

   $x\left[\begin{array}{c}1\\ 0\\ 2\end{array}\right]+y\left[\begin{array}{c}2\\ 1\\ 0\end{array}\right]+z\left[\begin{array}{c}-1\\ 1\\ 0\end{array}\right]$

2. $\left[\begin{array}{c}2z-y+x\\ x-y+z\\ y+2x\end{array}\right]$

   $x\left[\begin{array}{c}1\\ 1\\ 2\end{array}\right]+y\left[\begin{array}{c}-1\\ -1\\ 1\end{array}\right]+z\left[\begin{array}{c}2\\ 1\\ 0\end{array}\right]$