The key idea of this problem is to realize that the plane
contains
linearly independent vectors; For example,
.
Therefore, if
is a linear transformation that maps
into
,
in fact sends all of
to
. 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 }](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/f6e996a37798ecc80e0bde5f6cccb7dd0fcce91c)
for some constants
![{\displaystyle a}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
,
![{\displaystyle b}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3)
, and
![{\displaystyle c}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455)
.
On the other hand, we also need to ensure that at least one point of
is not mapped to the zero vector: otherwise the image of
under
would simply be the zero vector instead of all of
. In other words, at least one of
,
, and
is non-zero.
Therefore, the matrix of
must be of the form
![{\displaystyle T=\left[{\begin{array}{ccc}a&b&c\\a&b&c\\a&b&c\end{array}}\right],}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/d1b97cbd347e8a6350574f3badbac1a457bb4885)
where at least one of
![{\displaystyle a}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc)
,
![{\displaystyle b}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3)
, and
![{\displaystyle c}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455)
is non-zero.