Let $\mathbf{V}$ be $\mathbb{R}^5$ with the usual Euclidean inner product, and let $\mathbf{W}$ be the subspace of $\mathbf{V}$ spanned by the vectors $\mathbf{v}_1$, $\mathbf{v}_2$, $\mathbf{v}_3$, $\mathbf{v}_4$ where: $\begin{align*} \mathbf{v}_1&=[1,3,1,-2,3],\\\mathbf{v}_2&=[1,4,3,-1,-4],\\ \mathbf{v}_3&=[2,3,-4,-7,-3],\\\text{ and }\quad\mathbf{v}_4&=[3,8,1,-7,-8].\end{align*}$
- Find a basis for $\mathbf{W}$.
- Find an orthogonal basis for $\mathbf{W}$.
- Find an orthonormal basis for $\mathbf{W}$.
- Let vector $\mathbf{u}=[3,8,1,-7,-8]$. Is $\mathbf{u}$ in $\mathbf{W}$ or not? If it is, find the components of $\mathbf{u}$ with respect to the orthonormal basis found in 3.
I do know that $\mathbf{v}_1$, $\mathbf{v}_2$, $\mathbf{v}_3$, $\mathbf{v}_4$ do span $\mathbf{W}$.