Full solution

Q. Consider this matrix transformation:

\newline

\left[\begin{array}{ccc} -1 & 1 & 3 \\ 0 & 0 & 3 \\ -1 & 0 & 3 \end{array}\right]

\newline

What is the image of

\left[\begin{array}{c} 2 \\ 5 \\ 3 \end{array}\right]

under this transformation?

Define Transformation Domain: To find the image of the matrix transformation, we need to determine the set of all possible outputs (vectors) that can be obtained by applying the matrix to all vectors in the domain (which is typically $\mathbb{R}^n$ where $n$ is the number of columns of the matrix). In this case, the matrix has $3$ columns, so we are looking at the transformation from $\mathbb{R}^3$ to $\mathbb{R}^3$ .
Determine Image Basis: The image of a matrix transformation is spanned by the columns of the matrix. So, we need to look at the columns of the matrix $\begin{bmatrix} -1 & 1 & 3 \ 0 & 0 & 3 \ -1 & 0 & 3 \end{bmatrix}$ and determine if they are linearly independent to find the basis for the image.
Check Linear Independence: The columns of the matrix are: $\newline$ Column $1$ : $[-1, 0, -1]$ $\newline$ Column $2$ : $[1, 0, 0]$ $\newline$ Column $3$ : $[3, 3, 3]$ $\newline$ We can see that Column $3$ is a multiple of Column $2$ (Column $3$ = $3 \times$ Column $2$ ), which means they are not linearly independent.
Ignore Dependent Column: Since Column $3$ is a multiple of Column $2$ , we can ignore it when determining the basis for the image. The remaining columns are: $\newline$ Column $1$ : $[-1, 0, -1]$ $\newline$ Column $2$ : $[1, 0, 0]$ $\newline$ These two columns are linearly independent because there is no scalar multiple that can turn one into the other.
Find Image Span: The image of the matrix transformation is the span of the linearly independent columns. Therefore, the image is the plane spanned by the vectors $[-1, 0, -1]$ and $[1, 0, 0]$ .
Express Vector in Image: To describe the image more precisely, we can express any vector in the image as a linear combination of the basis vectors: $v = a[-1, 0, -1] + b[1, 0, 0]$ for all real numbers $a$ and $b$ .