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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: $$\begin{align*} \mathbf{v}_1&=[1,3,1,-2,3],\\\mathbf{v}_2&=[1,4,3,-1,-4],\\ \mathbf{v}_3&=[2,3,-4,-7,-3],\\\text{ and }\quad\mathbf{v}_4&=[3,8,1,-7,-8].\end{align*}$$

  1. Find a basis for $\mathbf{W}$.
  2. Find an orthogonal basis for $\mathbf{W}$.
  3. Find an orthonormal basis for $\mathbf{W}$.
  4. 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 $\mathbf{W}$.

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Well... of course they do! $\mathbf{W}$ is defined to be the subspace spanned by those vectors, so of course they span $\mathbf{W}$. Do you know how to extract a basis from a spanning set? (HINT: start getting rid of vectors that are linear combinations of vectors you already have). Do you know the Gram-Schmidt orthonormalization process? – Arturo Magidin Dec 8 '11 at 5:59

I do not attempt to give a full answer - this is a standard question, solved with standard techniques which you should familiarize yourself with.

  1. Use Gaussian Elimination on the matrix containing the given vectors.
  2. Use Gram-Schmidt.
  3. Ditto.
  4. Solve a linear equation system with $u$ being the right-hand side and the coefficients of the system given by the basis of 3.
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To find a basis you can put the 4 vectors as rows of a matrix, you can make elementary operations on the rows , and when you get an "echelon form" the nonzero rows are a basis. To get an orthonormal basis apply Gram-Schmidt.

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