Basis of Kernel of Linear Transformation

Question

Determine a basis of $ker(T)$ and $T(R^5)$ for a linear map $T : R^5 → R^3$ whose matrix (relative to the standard basis) equals

$\begin{bmatrix}1 & 0 & 0 & 5 & 9\\ 0 & 1 & -1 & -3 & 2\\ 0 &2 & -2 & -6 & 4 \end{bmatrix}$

To find the basis for kernel, I just let the matrix time a vector [v1,...,v5] s.t. it equals 0 and solve for the basis. For $T(R^5)$ basis, do I just reduce the matrix to rref and find the basis of the rref?

In addition, what does it mean "(relative to the standard basis)" in the question?

Answer 1

Relative to the standard basis just means we don't have to use a change of basis matrix to compute the kernel. So we get $\begin{bmatrix}1 & 0 & 0 & 5 & 9\\ 0 & 1 & -1 & -3 & 2\\ 0 &2 & -2 & -6 & 4 \end{bmatrix}\to\begin{bmatrix}1 & 0 & 0 & 5 & 9\\ 0 & 1 & -1 & -3 & 2\\ 0 &0 & 0 & 0 & 0 \end{bmatrix}$. Now "back-substitute". Get $b-c-3d+2e=0,a+5d+9e=0$. So $\{(-9,-2,0,0,1),(-5,3,0,1,0),(0,1,1,0,0)\}$ is a basis for the kernel.

Answer 2

You find the kernel if you evaluate the augmented matrix

\begin{pmatrix} 1 & 0 & 0 & 5 & 9 &\bigm| & 0 \\ 0 & 1 & -1 & -3 & 2&\bigm| & 0 \\ 0 & 2 & -2 & -6 & 4 &\bigm| & 0 \end{pmatrix}

(this is equivalent to solving $Ax = 0$).

The remainining linearly independed row vectors generate the kernel.

Relative to the standard basis means that your matrix has been evaluated on the standard basis $e_1,e_2,...,e_5$ of $\mathbb{R}^5$. In other words: the columns of your matrix are the images of each standard basis vector

$$T(e_1) = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, T(e_2) = \begin{pmatrix} 0\\ 1 \\ 2 \end{pmatrix},... $$

and so on and so forth.

Answer 3

The 3 row is a multiple of the second row. We have 2 independent row vectors. The dimension of the kernel will be 3.

$\begin{bmatrix} 0\\1\\1\\0\\0\end{bmatrix}$ is in the kernel.

We need two more.

Consider, $\begin{bmatrix} -5\\u\\v\\1\\0\end{bmatrix},\begin{bmatrix} -9\\w\\x\\0\\1\end{bmatrix}$

We know that whatever the values of $u,v,w,x$ are, they will first row will multiply to $0.$ Find values that make $0$ when multiplied by the other rows.

Clearly these 3 vectors are independent.