Column picture for representing a system of equations

[vanmaiden]

I've been dabbling with linear algebra lately and on the MIT OCW course for linear algebra, the professor talks about a column representation of a system of linear equations. For example, you teaches you to represent a

2x + 3y = 4
5x + 7y = 9

as a

x [$\stackrel{2}{5}$] + y [$\stackrel{3}{7}$] = [$\stackrel{4}{9}$]

The professor says one can perform this alternate way to solve a SoE, but what's the logic that allowed people to discover this new method?

 

[HallsofIvy]

Pretty much basic linear algebra. The two operations we have in a vector space are "addition of vectors" and "scalar multiplication". The "$x\begin{bmatrix}2 \\ 5\end{bmatrix}= \begin{bmatrix}2x \\ 5x\end{bmatrix}$" and "$y\begin{bmatrix}3 \\ 7\end{bmatrix}= \begin{bmatrix}3y \\ 7y\end{bmatrix}$" are "scalar multiplications" while $\begin{bmatrix}2x \\ 5x\end{bmatrix}+ \begin{bmatrix}3y \\ 7y\end{bmatrix}= \begin{bmatrix}2x+ 3y\\ 5x+ 7y\end{bmatrix}$ is "addition of vectors". Finally, the definition of "equality" for vectors tells us that $\begin{bmatrix}2x+ 3y \\ 5x+ 7y\end{bmatrix}= \begin{bmatrix}4 \\ 9\end{bmatrix}$ is the same as "2x+ 3y= 4" and "5x+ 7y= 9".