Science:Math Exam Resources/Courses/MATH152/April 2017/Question A 30/Solution 1

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 $$T\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]=a\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right],T\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]=b\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right],T\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]=c\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right],$$ 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.