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I'm not entirely sure how to go about answering this question about vectors. Any advice/help is appreciated.
Write the vector $\displaystyle a =\begin{bmatrix}3\\-1\\7\end{bmatrix}$ as a linear combination of the set of orthonormal basis vectors $\displaystyle\left\{\begin{bmatrix}1/\sqrt2\\0\\1/-\sqrt2\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix},\begin{bmatrix}1/\sqrt2\\0\\1/\sqrt2\end{bmatrix}\right\}$ in $\mathbb R^3.$
$$-2\sqrt{2} \begin{bmatrix}1/\sqrt2\\0\\1/-\sqrt2\end{bmatrix}-\begin{bmatrix}0\\1\\0\end{bmatrix}+5\sqrt{2}\begin{bmatrix}1/\sqrt2\\0\\1/\sqrt2\end{bmatrix}$$
Hint: If $\{u_1,\dots,u_n\}$ for an orthonormal basis for the space $X$, then any vector $x\in X$ can be written as$$x=\sum_{i=1}^n \langle x,u_i\rangle u_i$$
Let's call the vector you are interested in $\mathbf{x}$ and the basis vectors $\mathbf{v_1, v_2, v_3}$.
We know $\mathbf{x} = a\mathbf{v_1} + b\mathbf{v_2} + c\mathbf{v_3}$ for some $a, b, c \in \mathbb{R}$ and we wish to find those constants. Since the basis is orthonormal, each constant will simply be the value of $\mathbf{x} \cdot \mathbf{v_i}$. This is basically just performing a projection of the vector onto each of the three basis vectors.
