Finding out if a polynomial corresponds to a linar transformaion

**Sherina** · March 13th 2011, 04:06 PM

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?

**Drexel28** · March 13th 2011, 09:01 PM

Originally Posted by Sherina

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 $P_2$ there is a natural isomorphism $\phi$ between $P_2$ and $\mathbb{R}^3$ by $a_0+a_1x+a_2x^2\mapsto (a_0,a_1,a_2)$ and a similar one, call it $\varphi$ , between $P_3$ and $\mathbb{R}^4$ . If $M$ were a linear transformation then so would $\varphi^{-1}\circ M\circ \phi:\mathbb{R}^3\to\mathbb{R}^4$ . So, is $M$ a linear transformation?

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