A matrix can be regarded as a representation of a linear function from one vector space to another, with the condition that the basis vectors (and their order) are specified for both spaces.
Therefore a column matrix must necessarily be a representation of a vector of a space in relation to that space basis.
For example, let $\vec{u}=4\vec{e_1} + 1\vec{e_2}$. In matrix notation we have $\left[ \begin {matrix} 4 \\ 1 \\ \end {matrix} \right] = 4 \left[ \begin {matrix} 1 \\ 0 \\ \end {matrix} \right] + 1 \left[ \begin {matrix} 0 \\ 1 \\ \end {matrix} \right] $.
Let's assume the basis ($\vec{e_1}, \vec{e_2}$) looks like the vectors shown in the figure (The basis vectors and their linear combination).
My question is: what is the dot product of the basis vectors?
According to the matrix notation it should be zero: $ \left[ \begin {matrix} 1 \\ 0 \\ \end {matrix} \right] \cdot \left[ \begin {matrix} 0 \\ 1 \\ \end {matrix} \right] =0 $. But, according to the figure the dot product should not be zero as both vectors are clearly not orthogonal.
Obviously I am missing something. Any help would be appreciated.