# Image (mathematics)

### From Conservapedia

In mathematics, the image of a linear transformation is its range: all possible values generated by the transformation. A matrix A, which is an expression of a function, has an image denoted by im(A).

If the rows (or columns, equivalently) of a matrix A are linearly independent, then the image of that transformation is the entire space it is applied to.

## Examples

### Example 1

Consider all the points (vectors) in the plane, ie, (x,y) acting under the transformation

$\mathbf{A} = \begin{pmatrix} -1 & 1 \\ 1 & 0 \end{pmatrix}$.

We can be sure that the image of this transformation is the entire plane, because for any point

$\begin{pmatrix} x \\ y \end{pmatrix}$

in the plane, there is another vector in the plane

$\begin{pmatrix} y \\ x+y \end{pmatrix}$

such that

$\begin{pmatrix} -1 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix}y \\x+y \end{pmatrix} = \begin{pmatrix}x \\y \end{pmatrix}$.

### Example 2

We continue to work in the plane, but now we examine the matrix

$\mathbf{A} = \begin{pmatrix} 3 & 1 \\ -6 & -2 \end{pmatrix}$.

Now if we examine how this matrix acts on an arbitary point (a,b), we find that point is carried to

$\mathbf{A} = \begin{pmatrix}3 & 1 \\-6 & -2 \end{pmatrix}\begin{pmatrix}a \\b \end{pmatrix} = \begin{pmatrix}3a+b \\-6a-2b \end{pmatrix} = \begin{pmatrix}1 \\-3 \end{pmatrix}(a+b)$,

in other words, all points in the plane are carried to the line $y=-3x \$.

We write

$im(\mathbf{A}) = \begin{pmatrix}1 \\-3 \end{pmatrix}t, \forall t\in\mathbb{R} \$.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 8, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.898945152759552, "perplexity_flag": "middle"}