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 $$T{\begin{bmatrix}1\\0\\0\end{bmatrix}}=a{\begin{bmatrix}1\\1\\1\end{bmatrix}},\quad T{\begin{bmatrix}0\\1\\0\end{bmatrix}}=b{\begin{bmatrix}1\\1\\1\end{bmatrix}},\quad T{\begin{bmatrix}0\\0\\1\end{bmatrix}}=c{\begin{bmatrix}1\\1\\1\end{bmatrix}},\quad$$ 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.