by Guest » Fri Apr 16, 2021 12:21 pm
The "kernel" of a linear transformation, T, is the subspace of all vectors, v, such that Tv= 0.
Here, T is the linear transformation that maps the 2 by 2 matrix, [tex]\begin{pmatrix}x & y \\ z & t \end{pmatrix}[/tex] to the three-vector [tex]\begin{pmatrix} x+ t & y- x & z\end{pmatrix}[/tex]. If the matrix is in the kernel of T, then we must have x+ t= 0, y- x= 0, and z= 0. From x+ t= 0, t= -x. From y- x= 0, y= x. So any matrix in the kernel of T must be of the form [tex]\begin{pmatrix}x & x \\ 0 & -x\end{pmatrix}= x\begin{pmatrix}1 & 1 \\ 0 & -1\end{pmatrix}[/tex].
What is the dimension of that space?