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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.