Image (mathematics)

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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

.

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

in the plane, there is another vector in the plane

such that

.

Example 2

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

.

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

,

in other words, all points in the plane are carried to the line .

We write

.