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.


Example 1

Consider all the points (vectors) in the plane, i.e., (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 arbitrary 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