I'm stuck while trying to understand the Hadamard Gate in a more linear algebra understanding. (I understand the algebraic way). This is because I want to program a simulation of a quantum computer. To apply a gate you multiply each ket by the unitary matrix.
So, the Hadamard gate maps the state $\alpha |0\rangle + \beta|1\rangle$ to $\frac{\alpha}{\sqrt{2}}\begin{bmatrix}1\\1\\\end{bmatrix}+\frac{\beta}{\sqrt{2}}\begin{bmatrix}1\\-1\\\end{bmatrix}$ right? But the outputs are not basis vectors. I know this is hard to understand but how do I make it back to the basis vector form so I can write it in ket form. I know you can do it algebraically but how to in linear algebra?
So put it in the form: $$\alpha\begin{bmatrix}1\\0\\\end{bmatrix}+\beta\begin{bmatrix}0\\1\\\end{bmatrix}$$
If you understand what I mean, can you explain it in a general matrix form?