TITLE: Meaning of vector equations
QUESTION [5 upvotes]: I'm taking a linear algebra class now, and I was introduced to vector equations. Consider the system 
$$
\left\{ \begin{array}{rcl}
x-4y &=& 8\\
2x+3y &=& 6\\
\end{array}
\right.$$
I want to understand why I can factor out the x and y variables to create the vector equation $$x\begin{bmatrix} 1 \\ 2 \end{bmatrix}+y\begin{bmatrix} -4 \\ 3 \end{bmatrix}=\begin{bmatrix} 8 \\ 6 \end{bmatrix}$$
are $x$ and $y$ scalars here? Is the column vector $\begin{bmatrix} 1 \\ 2 \end{bmatrix}$ the same as the vector $\left< 1, 2\right>$? And lastly, how does this notation help me solve for solutions?

REPLY [4 votes]: I agree with the other answers that the matrix form is the more standard way of viewing the system of linear equations.
But the vector way of viewing things can be thought of as in a 2-d diagram, as we are dealing with two-dimensional vectors.
Clearly $\begin{bmatrix} 1 \\ 2 \end{bmatrix}$ and $\begin{bmatrix} -4 \\ 3 \end{bmatrix}$ are linearly independent, as one is not a scalar multiple of the other.
Therefore, the equation:
$$x\begin{bmatrix} 1 \\ 2 \end{bmatrix}+y\begin{bmatrix} -4 \\ 3 \end{bmatrix}=\begin{bmatrix} 8 \\ 6 \end{bmatrix}$$
can be thought of as, "What unique linear combination of the $2-$d vectors $\begin{bmatrix} 1 \\ 2 \end{bmatrix}$ and $\begin{bmatrix} -4 \\ 3 \end{bmatrix}$ is equal to $\begin{bmatrix} 8 \\ 6 \end{bmatrix},$ and personally I kind of find this uniqueness kind of "interesting" or "nice", I suppose.