The Image of a Linear Transformation

The image of a linear transformation
$T$
operating on a set
$S$
is the set of elements
$T(x) \in T(S)S$
i.e.e the set of elements of the codomain that are the image of some element of


Example. Differentiation is linear. We can define a linear transformation on the set of polynomials of degree 2.
$\frac{d}{dx}(1)=0,\frac{d}{dx}(x)=1, \frac{d}{dx}(x^2)=2x$
.
We represent
$1.x.x^2$
by the vectors
$\begin{pmatrix}1\\0\\0\end{pmatrix}, \begin{pmatrix}0\\1\\0\end{pmatrix}, \begin{pmatrix}0\\0\\1\end{pmatrix}$
Hence
$\frac{d}{dx}(a+bx+cx^2)=b+2cx$
.
The columns of the matrix representing
$T$
can be found by differentiating
$1.x.x^2$
in turn and representing the results as vectors.
The matrix representing th linear transformation is
$\left( \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 &2 \end{array} \right)$
.
The polynomial
$2+3x+5x^2$
is represented by the vector
$\begin{pmatrix}2\\3\\5\end{pmatrix}$

$T(2+3x+5x^2)= \left( \begin{array}{cc} 0 & 1 & 0 \\ 0 & 0 & 2 \end{array} \right) \begin{pmatrix}2\\3\\5\end{pmatrix}=\begin{pmatrix}3\\10\\0\end{pmatrix}$
which returns the polynomial
$3+10x$
.
The image of this transformation is the set of polynomials of degree 2, since the differentiation reduces the degree of polynomials by 1..
The image of a transformation has certain properties. The image of a transformation is a subspace of the codomain.
1.
$T(0)=0$

2. If
$T(x),T(y) \in im(T) \rightarrow T(\alpha x+ \beta y)= \alpha T(x) + \beta T(y) \in im(T)$

The dimension of the image as a subspace is always less than or equal to the dimension of the domain, considered as a subspace.