Meaning of vector equations 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?
 A: The fact is that any linear system in cartesian form can also be expressed in matrix form $Au=b$ and the product $Au$ can also be viewed as the linear combination of the columns of matrix $A$ by the components of the vector $u$.
$$\overbrace{\boxed{\left\{ \begin{array}{rcl}
a_{11}x+a_{12}y &=& b_1\\
a_{21}x+a_{22}y &=& b_2\\
\end{array}
\right.}}^{\text{cartesian form}}
\,\equiv\, 
\overbrace{\boxed{\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}
\begin{bmatrix} x \\ y \end{bmatrix}
=
\begin{bmatrix} b_1 \\ b_2 \end{bmatrix}}}^{\text{matrix form}} 
\,\equiv\, 
\overbrace{\boxed{x\begin{bmatrix} a_{11} \\ a_{21} \end{bmatrix}+y\begin{bmatrix} a_{12} \\ a_{22} \end{bmatrix}=\begin{bmatrix} b_1 \\ b_2 \end{bmatrix}}}^{\text{vector form}}$$
In some case this interpretation (i.e. solution as a combination of matrix column vectors) can be very useful. In this particular case it seems not much enlightening  to find the solution.
As reference I suggest to take a look to the first lesson of the famous Linear Algebra course by Prof. Gilbert Strang.
A: Yes $x,y$ are scalars.
Yes the column vectors are the same as you mentioned.
This notation is classical in linear algebra because you can go ahead and write the above as
$$\begin{bmatrix} 1 & -4 \\ 2 & 3 \end{bmatrix}
\begin{bmatrix} x \\ y \end{bmatrix}
=
\begin{bmatrix} 8 \\ 6 \end{bmatrix}$$
where now all you have to do is "invert" the matrix to obtain $x,y$.
A: 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.
