Science:Math Exam Resources/Courses/MATH221/December 2009/Question 04

MATH221 December 2009

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Question 04
Link to the exam for the picture Let $W$ be the subspace of $R^{3}$ spanned by ${\vec {w}}_{1}$ and ${\vec {w}}_{2}$ where ${\vec {w}}_{1}=[1,0,-1]$ and ${\vec {w}}_{2}=[-1,2,-1]$ Let T : $R^{3}->R^{3}$ be the reflection across W, find the standard matrix T. INSERT PICTURE HERE

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

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Solution

Whoops, no solution has been written yet. Be the first to submit a suggestion. A vector normal to the plain is given by ${\vec {w}}_{1}\times {\vec {w}}_{2}$ which is ${\vec {w_{1}}}\times {\vec {w_{2}}}={\begin{bmatrix}-2\\0\\2\end{bmatrix}}$ To find the T matrix we need to find how the basis vectors $e_{1}=[1,0,0],e_{2}=[0,1,0],e_{3}=[0,0,1]$ are transformed. It is easy to see that $e_{2}=[0,1,0]$ gets transformed to itself because it lies on the plane. $e_{1}$ and $e_{3}$ are more difficult. For $e_{1}$, imagine that we were looking directly on the x,z plane, the plane would then be squished into the line x = z. Then it is clear that the vector $e_{1}=[1,0,0]$ transforms into $[0,0,1]$. Similarly, we can say that $e_{3}=[0,0,1]$ transforms into $[1,0,0]$ Hence, our transformation matrix is $T={\begin{bmatrix}0&0&1\\0&1&0\\1&0&0\end{bmatrix}}$