Find the kernel and range of the following transformation If the given transformation is
$$T(\begin{bmatrix}V_1 \\ V_2 \\ V_3\end{bmatrix})=\begin{bmatrix}V_1 \\V_2\end{bmatrix}$$
Find the column space and null space for the given transformation
 A: Before seeing the answer, please see this source.
Given,
$$
T\cdot \begin{bmatrix}
v_1\\
v_2\\
v_3\\
\end{bmatrix} = \begin{bmatrix}v_1\\ v_2\\ \end{bmatrix}
$$
Calculating nullspace is easy.
$$
T\cdot \begin{bmatrix}
0\\
0\\
k\\
\end{bmatrix} = \begin{bmatrix}0\\ 0\\ \end{bmatrix}
$$
Therefore $Nullspace(T) = span\left\{\begin{bmatrix} 0 \\ 0\\ 1\\ \end{bmatrix}\right\}$
To get the column space or range we need to find the matrix behind this transformation. We gonna look at what $T$ does to each basis vector.
$$
T\cdot \begin{bmatrix}
1\\
0\\
0\\
\end{bmatrix} = \begin{bmatrix}1\\ 0\\ \end{bmatrix}
$$
$$
T\cdot \begin{bmatrix}
0\\
1\\
0\\
\end{bmatrix} = \begin{bmatrix}0\\ 1\\ \end{bmatrix}
$$
$$
T\cdot \begin{bmatrix}
0\\
0\\
1\\
\end{bmatrix} = \begin{bmatrix}0\\ 0\\ \end{bmatrix}
$$
Therefore the matrix looks like $T = \begin{bmatrix}1 & 0& 0 \\ 0 & 1& 0\end{bmatrix}$
Above $T$ is already is in reduced row-echelon form. Therefore
$$
Col(T) = span\left\{ \begin{bmatrix}1 \\0 \end{bmatrix}, \begin{bmatrix}0 \\1 \end{bmatrix} \right\}
$$
