Hello,
the notation with the matrix is simply another way to express the same facts. If you have the components of two vectors, then you can also transfer these into a matrix, that is exactly what is done there. Multiply once the matrix multiplied by the vector (x, y)^T, you will see that the same comes out. The notation as a linear system of equations is much clearer here, at least with several vectors.
Two vectors are linear dependent, If one can write one vector in terms of another, here in that little example you can obtain, that it is possible to rewrite [MATH]\vec{a}_2[/MATH] in terms of [MATH]\vec{a}_1[/MATH], just by multilpy 2 times [MATH]\vec{a}_2[/MATH]
[MATH]\vec{a}_1 = \begin{pmatrix} 4 \\ 6 \end{pmatrix}, \vec{a}_2 = \begin{pmatrix} 2 \\ 3 \end{pmatrix}[/MATH]See:
[MATH]\vec{a}_1 = \begin{pmatrix} 4 \\ 6 \end{pmatrix} = 2\begin{pmatrix} 2 \\ 3 \end{pmatrix} = 2\cdot \vec{a}_2[/MATH]
