Science:Math Exam Resources/Courses/MATH152/April 2017/Question A 30

MATH152 April 2017

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Question A 30
Let $P$ be a plane and $L$ a line in $\mathbb {R} ^{3}$ given by equations ${\begin{aligned}&P:x+y+z=1\\&L:u{\begin{bmatrix}1\\1\\1\end{bmatrix}}\quad u\in \mathbb {R} .\end{aligned}}$ Find all linear transformations such that the image of $P$ is $L$ (that is the set of all outputs is the line $L$ when all points on the plane $P$ are taken as inputs).

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
The plane $P$ does not pass through the origin, it follows that $P$ contains three points that, when viewed as vectors in $\mathbb {R} ^{3}$, are linearly independent.

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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 key idea of this problem is to realize that the plane $P$ contains $3$ linearly independent vectors; For example, $(1,0,0)^{T},(0,1,0)^{T},(0,0,1)^{T}$. Therefore, if $T$ is a linear transformation that maps $P$ into $L$, $T$ in fact sends all of $\mathbb {R} ^{3}$ to $L$. In particular, we have $$T{\begin{bmatrix}1\\0\\0\end{bmatrix}}=a{\begin{bmatrix}1\\1\\1\end{bmatrix}},\quad T{\begin{bmatrix}0\\1\\0\end{bmatrix}}=b{\begin{bmatrix}1\\1\\1\end{bmatrix}},\quad T{\begin{bmatrix}0\\0\\1\end{bmatrix}}=c{\begin{bmatrix}1\\1\\1\end{bmatrix}},\quad$$ for some constants $a$, $b$, and $c$. On the other hand, we also need to ensure that at least one point of $P$ is not mapped to the zero vector: otherwise the image of $P$ under $T$ would simply be the zero vector instead of all of $L$. In other words, at least one of $a$, $b$, and $c$ is non-zero. Therefore, the matrix of $T$ must be of the form $$T=\left[{\begin{array}{ccc}a&b&c\\a&b&c\\a&b&c\end{array}}\right],$$ where at least one of $a$, $b$, and $c$ is non-zero.