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