Science:Math Exam Resources/Courses/MATH221/December 2011/Question 06 (a)

MATH221 December 2011

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[hide] Question 06 (a)
Consider the vectors $v_{1}=\left({\begin{array}{c}1/2\\1/2\\1/2\\1/2\end{array}}\right),\quad v_{2}=\left({\begin{array}{c}1/2\\1/2\\-1/2\\-1/2\end{array}}\right),$ $v_{3}=\left({\begin{array}{c}1/2\\-1/2\\1/2\\-1/2\end{array}}\right),\quad v_{4}=\left({\begin{array}{c}1/2\\-1/2\\-1/2\\1/2\end{array}}\right)$ a) Check that v1, v2, v3, v4 is an orthogonal basis of R4.

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

[show] Hint 1
Check whether the vectors are pairwise orthogonal.

[show] Hint 2
The four vectors will form a basis of $\mathbb {R} ^{4}$ if they are linearly independent.

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[show] Solution

First we must determine whether the vectors are orthogonal. We know that two vectors are orthogonal if their dot product is zero. Consider $v_{1}\cdot v_{2}$. This is equal to $(1/2)(1/2)+(1/2)(1/2)+(1/2)(-1/2)+(1/2)(-1/2)=0.$ All other dot products are calculated in the same way and all vanish. Hence $\{v_{1},v_{2},v_{3},v_{4}\}$ is an orthogonal set. Since an orthogonal set of nonzero vectors must be linearly independent, and a set of 4 linearly independent vectors in $\mathbb {R} ^{4}$ must be a basis of $\mathbb {R} ^{4}$, we conclude that $\{v_{1},v_{2},v_{3},v_{4}\}$ is an orthogonal basis of $\mathbb {R} ^{4}$.