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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 $$\mathbf v^{(0)} = \begin{bmatrix}\frac 12\\\frac 12\\\frac 12\\\frac 12\end{bmatrix}, \mathbf v^{(1)} = \begin{bmatrix}\frac 12\\\frac 12\\-\frac 12\\-\frac 12\end{bmatrix}, \mathbf v^{(2)} = \begin{bmatrix}\frac 12\\-\frac 12\\\frac 12\\-\frac 12\end{bmatrix}, \mathbf v^{(3)} = \begin{bmatrix}\frac 12\\-\frac 12\\-\frac 12\\\frac 12\end{bmatrix}, \quad $$

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This is actually a math problem, but it is related to signal processing via the concept of signal space. Since it's a homework-style problem you'll only get hints that should help you solve the problem on your own.

  1. verify that the given vectors are orthonormal.
  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., $$\mathbf{y}=\sum_{i=0}^3(\mathbf{y}\cdot \mathbf{v}^{(i)})\mathbf{v}^{(i)}$$

I don't understand your question about how to compute the inner product. If you know (or look up) the definition of the inner product then you should be able to compute it.

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