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, 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


x  \\
y  \end{pmatrix}

in the plane, there is another vector in the plane


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} \ .

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