Science:Math Exam Resources/Courses/MATH152/April 2012/Question 03 (a)

MATH152 April 2012

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Question 03 (a)

Let A be the matrix for the linear transformation T. Suppose we know that $\mathbf {v} _{1}={\begin{bmatrix}1\\1\\1\end{bmatrix}}$, $\mathbf {v} _{2}={\begin{bmatrix}0\\1\\1\end{bmatrix}}$, $\mathbf {v} _{3}={\begin{bmatrix}1\\0\\1\end{bmatrix}}$ are eigenvectors of A associated to the eigenvalues $\lambda =1$, $\lambda =2$, and $\lambda =3$ respectively. Express $\mathbf {e} _{1}={\begin{bmatrix}1\\0\\0\end{bmatrix}}$, $\mathbf {e} _{2}={\begin{bmatrix}0\\1\\0\end{bmatrix}}$, and $\mathbf {e} _{3}={\begin{bmatrix}0\\0\\1\end{bmatrix}}$ as linear combinations of $\mathbf {v} _{1}$, $\mathbf {v} _{2}$ and $\mathbf {v} _{3}$.

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
Expressing a vector $\mathbf {x}$ as a linear combination of three vectors $\mathbf {u} ,\mathbf {v} ,\mathbf {w}$ means finding coefficients a, b, c such that $\mathbf {x} =a\mathbf {u} +b\mathbf {v} +c\mathbf {w}$

Checking a solution serves two purposes: helping you if, after having used the hint, 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

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Let's start with $\mathbf {e} _{1}$. We could solve these problems by inspection but let's, at least for this case, show how one could go about solving this schematically. We want to find a, b, c such that ${\begin{aligned}\mathbf {e} _{1}=a\mathbf {v} _{1}+b\mathbf {v} _{2}+c\mathbf {v} _{3}\\{\begin{bmatrix}1\\0\\0\end{bmatrix}}=a{\begin{bmatrix}1\\1\\1\\\end{bmatrix}}+b{\begin{bmatrix}0\\1\\1\end{bmatrix}}+c{\begin{bmatrix}1\\0\\1\end{bmatrix}}.\end{aligned}}$ Writing the above in matrix form we get ${\begin{bmatrix}1&0&1\\1&1&0\\1&1&1\end{bmatrix}}{\begin{bmatrix}a\\b\\c\end{bmatrix}}={\begin{bmatrix}1\\0\\0\end{bmatrix}}.$ In general we will always get a matrix with columns that are the vectors we are trying to combine. If we row-reduce this matrix (try it for yourself!) we will get $\left[{\begin{array}{ccc|c}1&0&0&1\\0&1&0&-1\\0&0&1&0\end{array}}\right]$ Hence a = 1, b = -1, c = 0, and so finally we can write $\mathbf {e} _{1}=\mathbf {v} _{1}-\mathbf {v} _{2}.$ We want to emphasize that it is perfectly fine to inspect the solution visually, however this technique will always work if you are stuck. For the other vectors we will write down the answers by inspection. For $\mathbf {e} _{2}$, we have $\mathbf {e} _{2}=\mathbf {v} _{1}-\mathbf {v} _{3}$ and for $\mathbf {e} _{3}$ we have $\mathbf {e} _{3}=-\mathbf {v} _{1}+\mathbf {v} _{2}+\mathbf {v} _{3}.$

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