How can I find expansion coefficients of the y vector in a given basis? Consider the following vectors in $\mathbb R^4$:

$$\mathbf{v}^{(0)}=\begin{bmatrix}\frac{1}{2}\\\frac{1}{2}\\\frac{1}{2}\\\frac{1}{2} \end{bmatrix} , \mathbf{v}^{(1)}=\begin{bmatrix}\frac{1}{2}\\\frac{1}{2}\\-\frac{1}{2}\\-\frac{1}{2} \end{bmatrix}, \mathbf{v}^{(2)}=\begin{bmatrix}\frac{1}{2}\\-\frac{1}{2}\\\frac{1}{2}\\-\frac{1}{2} \end{bmatrix} 
\mathbf{v}^{(3)}=\begin{bmatrix}\frac{1}{2}\\-\frac{1}{2}\\\frac{-1}{2}\\\frac{1}{2} \end{bmatrix}
$$

Let $$\mathbf{y}=\begin{bmatrix}0.5\\1.5\\-0.5\\0.5\\\end{bmatrix} \text,$$

what are the expansion coefficients of $\mathbf{y}$ in the basis $\{\mathbf{v}_0, \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$?
You need to find projection of $\vec{y}$ along unit vectors in the direction of each of the basis vectors $\mathbf v^{(i)}$. 

For finding unit vector in the direction of the vector, you just divide the vector by its magnitude.

And, for finding projection along a unit-vector, you just take the dot-product with the unit-vector.

So, the combined steps boil down to doing the following: $$&lt;\vec{y}, \frac{\mathbf v^{i}}{||\mathbf v^{i}||^{2}_{2}}&gt;$$

In your case the basis-vectors are already normalized, meaning they are already unit vectors, so just take the dot-product(inner product) of vector $\vec{y}$ with each of the basis vectors $\mathbf v^i$