# How can I find expansion coefficients of the a vector in a given basis?

How can I find the coefficient of the vector $$\mathbf y$$? And how can the inner product be done on these vectors?

Let $$\mathbf y = \begin{bmatrix}1\\2\\0\\1\end{bmatrix}$$

What are the expansion coefficients of $$\mathbf y$$ in the basis $$\left\{\mathbf v^{(0)}, \mathbf v^{(1)}, \mathbf v^{(2)}, \mathbf v^{(3)}\right\}$$ where $$\mathbf v^{(0)} = \begin{bmatrix}\frac 12\\\frac 12\\\frac 12\\\frac 12\end{bmatrix}, \mathbf v^{(1)} = \begin{bmatrix}\frac 12\\\frac 12\\-\frac 12\\-\frac 12\end{bmatrix}, \mathbf v^{(2)} = \begin{bmatrix}\frac 12\\-\frac 12\\\frac 12\\-\frac 12\end{bmatrix}, \mathbf v^{(3)} = \begin{bmatrix}\frac 12\\-\frac 12\\-\frac 12\\\frac 12\end{bmatrix}, \quad$$

2. if you've convinced yourself that the vectors are orthonormal, you can conclude that the expansion coefficients are simply the inner product of the given vector with the basis vectors, i.e., $$\mathbf{y}=\sum_{i=0}^3(\mathbf{y}\cdot \mathbf{v}^{(i)})\mathbf{v}^{(i)}$$