Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $\mathbf{W}$.

share|improve this question
2  
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
1  
What did you try? What do you know? Where did you fail? // This looks like homework. If it is, you should add the (homework) tag. –  Did Dec 8 '11 at 6:41
add comment

2 Answers

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.
share|improve this answer
add comment

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.