I'm having some trouble understanding the process of actually finding what $[T]_\beta ^\gamma$ is, given $2$ bases $\beta$ and $\gamma$. Here's an example:
Let $T$: $P_3(\mathbb{R})$ $\rightarrow$ $P_2(\mathbb{R})$ be the linear transformation defined by $T(f(x))$ = $f'(x)$. Let $\beta$ and $\gamma$ be the standard ordered bases for $P_3$ and $P_2$ (over $\mathbb{R}$), respectively. Find $[T]_\beta ^\gamma$.
What exactly is the step-by-step process of finding $[T]_\beta ^\gamma$? I know it's a very simple concept, but it's just not clicking with me.
Best Answer
Let $\beta=(\beta_1,\beta_2,\beta_3)$ and $\gamma=(\gamma_1,\gamma_2)$, where $\beta_1,\beta_2,\beta_3\in P_3(\Bbb{R})$ and $\gamma_1,\gamma_2\in P_2(\Bbb{R})$. The columns of the matrix $[T]_{\beta}^{\gamma}$ are the images of the basis vectors of $\beta$ under $T$, written on the basis $\gamma$. These are $$T(\beta_1)=\beta_1',\qquad T(\beta_2)=\beta_2',\qquad T(\beta_3)=\beta_3',$$ but to express them in terms of $\gamma_1$ and $\gamma_2$ you need to know (more about) what the bases $\beta$ and $\gamma$ are exactly. For example, on the standard bases $E_3=(1,x,x^2)$ and $E_2=(1,x)$ for $P_3(\Bbb{R})$ and $P_2(\Bbb{R})$ respectively, you have \begin{align*} T(1)=&\ 0&=0\cdot1+0\cdot x\\ T(x)=&\ 1&=1\cdot1+0\cdot x\\ T(x^2)=&\ 2x&=0\cdot1+2\cdot x \end{align*} so the matrix $[T]_{E_3}^{E_2}$ is given by $$[T]_{E_3}^{E_2}=\begin{pmatrix} 0&1&0\\0&0&2\end{pmatrix}$$