I'm learning Linear Transformations and I understand what is a linear transformation. Now I'm trying to look at an example question and I'm not really sure how the span is found. The question goes as follows:
Let $M_{2\times2}$ be the vector space of $2\times2$ matrices and let $R_{2}[x]$ be the space of quadratic polynomials. Consider
\begin{equation*}T: R_{2}[x] {\longrightarrow} M_{2\times2},
T(p) = \begin{bmatrix}
p(1)-p(2) & 0\\
0 & p(0) \\
\end{bmatrix}
\end{equation*}
Now, I want to find the Range of T, $R(T)$. I know that a basis for $R_{2}[x]$ is $B = \{1,x, x^2\}$ and therefore $R(T)$ can be found by finding the Span of the image of the basis elements under $T$.
\begin{equation*}
R(T) = span\{T(1), T(x), T(x^2)\}
\end{equation*}
This is where I get stuck. I'm not sure how to get the span from here on. The answer from the example in the book is as follows:
\begin{equation*}
R(T) = span\{\begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix}, \begin{bmatrix}-1 & 0 \\ 0 & 0\end{bmatrix}\}
\end{equation*}
Could someone explain in detail how the Span is determined? I realize that we have to replace $1, x$ and $x^2$ in $T(p)$ but I can't seem to get it right.
Best Answer
I assume everything is considered as a real vector space. The span is characterized as the set of linear combinations of elements. So the span of these matricies, call them $A=T(1),B=T(x)$, are the sums $rA+sB$ where $r,s \in \mathbb{R}$. This is equivalent to the set $$\bigg\{M\in \mathcal{M}_{2\times 2}: M=\begin{pmatrix} -s & 0 \\ 0 & r\\ \end{pmatrix}; r,s\in \mathbb{R}\bigg\}$$
Since every $r,s$ is possible, this is the set of all $2\times 2$ diagonal matrices.
If we note that $A=T(1)$, $B=T(x)$, $3B=T(x^2)$ we can further write for any polynomial given by $P(x)=a+bx+cx^2$ that $T(P(x))=aA+bB+3cB$ and write explicitly what this matrix is in the image.