Kernel of a linear transformation

Algebra

Kernel of a linear transformation

Postby Guest » Thu Apr 01, 2021 10:49 am

Let T: M (2,2) -> R3 given by (attached image), find the kernel of T and its dimension.
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Re: Kernel of a linear transformation

Postby 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?
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