Science:Math Exam Resources/Courses/MATH221/December 2008/Question 04 (a)

MATH221 December 2008

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Question 04 (a)
Let W be the plane spanned by the vectors $v={\begin{pmatrix}1\\1\\1\\1\end{pmatrix}}$ and $w={\begin{pmatrix}2\\0\\2\\0\end{pmatrix}}$. Verify that the vector $u={\begin{pmatrix}0\\1\\0\\1\end{pmatrix}}$ is in the subspace W and show that the vectors u and w form an orthogonal basis for W.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
What is the relation between orthogonal vectors and linear independence?

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The vector u can be written as $u=v-{\frac {1}{2}}w$, so indeed $u\in {\text{span}}\{v,w\}=W$. To show that u and w form an orthogonal basis for W, we first check (or see by observation) that their inner-product is zero, $\langle u,w\rangle =2(0)+1(0)+2(0)+1(0)=0$, so they are orthogonal. Now recall that in an inner-product space, orthogonal vectors are linearly independent, so u and w are linearly independent vectors in W. Moreover, u and w span W, precisely because $v=u+{\frac {1}{2}}w$, so every vector in W is an element in ${\text{span}}\{u,w\}$. Therefore u and w form a basis for W.

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