Science:Math Exam Resources/Courses/MATH152/April 2013/Question B 01 (d)

MATH152 April 2013

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Question B 01 (d)
Let $L$ be the line $3x=4y$. Find an eigenvector of the matrix A associated to the eigenvalue -1 where A was computed in part (b) and represents the reflection across the line L.

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.

Hint 1
Remember that the eigenvector(s), $\mathbf {v}$ corresponding to a particular eigenvalue $\displaystyle \lambda$ of a matrix $\displaystyle A$ satisfy $A\mathbf {v} =\lambda \mathbf {v} \quad \rightarrow \quad (A-\lambda I)\mathbf {v} =\mathbf {0}$ where I is the identity matrix.

Hint 2
For an alternative solution, think about which vector flips it's direction when reflected along a line?

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution 1

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. From the statement of the problem we know that $\lambda =-1$ is an eigenvalue of the matrix A. The corresponding eigenvector $\mathbf {v}$ satisfies the following equation $(A-\lambda I)\mathbf {v} =(A+I)\mathbf {v} =\mathbf {0}$ where I is the identity matrix. Solving for $\mathbf {v}$ gives: ${\begin{aligned}(A+I)\mathbf {v} &={\begin{bmatrix}{\frac {7}{25}}+1&{\frac {24}{25}}\\{\frac {24}{25}}&{\frac {-7}{25}}+1\end{bmatrix}}\mathbf {v} ={\frac {2}{25}}{\begin{bmatrix}16&12\\12&9\end{bmatrix}}\mathbf {v} =\mathbf {0} \quad \rightarrow \quad {\color {blue}\mathbf {v} ={\begin{bmatrix}3\\-4\end{bmatrix}}}.\end{aligned}}$ Any non-zero constant multiple of $\mathbf {v}$ is also an acceptable answer.

Solution 2

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Any vector perpendicular to the line of reflection will flip it's direction, and hence be an eigenvector with eigenvalue -1. To find the vector that is perpendicular to the line we rewrite 3x=4y in normal form $\langle 3,-4\rangle \cdot \langle x,y\rangle =0$ In other words, all point on the line 3x=4y are perpendicular to $\langle 3,-4\rangle$. Hence $\langle 3,-4\rangle$ must be an eigenvector with eigenvalue -1.

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