I'm not sure what this problem is asking me to do:
Consider the operation M:P2 -> P3 that takes p(x) in P2 to xp(x) in P3. Does this correspond to a linear transformation from R^3 to R^4? if so, what is its matrix?
I'm not sure what this problem is asking me to do:
Consider the operation M:P2 -> P3 that takes p(x) in P2 to xp(x) in P3. Does this correspond to a linear transformation from R^3 to R^4? if so, what is its matrix?
It's asking you for this. We know that for $\displaystyle P_2$ there is a natural isomorphism $\displaystyle \phi$ between $\displaystyle P_2$ and $\displaystyle \mathbb{R}^3$ by $\displaystyle a_0+a_1x+a_2x^2\mapsto (a_0,a_1,a_2)$ and a similar one, call it $\displaystyle \varphi$, between $\displaystyle P_3$ and $\displaystyle \mathbb{R}^4$. If $\displaystyle M$ were a linear transformation then so would $\displaystyle \varphi^{-1}\circ M\circ \phi:\mathbb{R}^3\to\mathbb{R}^4$. So, is $\displaystyle M$ a linear transformation?