# Calculate the image and a basis of the image (matrix)

What's the image of the matrix? What's the basis of the image? $M=\begin{pmatrix} -1 & 1 & 1\\ -2 & -3 & 6\\ 0 & -1 & 1 \end{pmatrix}$

First transposed the matrix:

$M^{T}=\begin{pmatrix} -1 & -2 & 0\\ 1 & -3 & -1\\ 1 & 6 & 1 \end{pmatrix}$

Now we use Gauss and get zero lines. Take the first line and add it to the third:

$M^{T}=\begin{pmatrix} -1 & -2 & 0\\ 0 & -5 & -1\\ 1 & 6 & 1 \end{pmatrix}$

Take the first line and add it to the third:

$M^{T}=\begin{pmatrix} -1 & -2 & 0\\ 0 & -5 & -1\\ 0 & 4 & 1 \end{pmatrix}$

Multiply the second line with $4$, multiply the third line with $5$, then add second line to third:

$M^{T}=\begin{pmatrix} -1 & -2 & 0\\ 0 & -20 & -4\\ 0 & 0 & 1 \end{pmatrix}$

Transpose back:

$M=\begin{pmatrix} -1 & 0 & 0\\ -2 & -20 & 0\\ 0 & -4 & 1 \end{pmatrix}$

The image of the matrix is $\text{Im(M)}= \text{span} \left ( \left\{ \begin{pmatrix} -1\\ -2\\ 0 \end{pmatrix}, \begin{pmatrix} 0\\ -20\\ 4 \end{pmatrix},\begin{pmatrix} 0\\ 0\\ 1 \end{pmatrix} \right\} \right)$

The basis of the image is $\left\{ \begin{pmatrix} -1\\ -2\\ 0 \end{pmatrix}, \begin{pmatrix} 0\\ -20\\ 4 \end{pmatrix},\begin{pmatrix} 0\\ 0\\ 1 \end{pmatrix} \right\}$

Please tell me if I did everything correctly? It's very important for me to know as I would do it like that in the exam :)

I hope it's correct and please also tell me if the notation is.

• Hi! Welcome to the Mathematics Stack Exchange! Congratulations on your first answer! $(+1)$ May 11, 2018 at 6:21