Exploring Linear Transformations on Basis Elements of P3(R)

[Butelle]

Hi

I am trying to do a math assignment and I am finding it really difficult.

Assume you have a linear transformation from T: P3(R) --> R4

What relevance is there to applying the transformation to the basis elements of P3(R), ie: T(1), T(x), T(x^2), T(x^3)? Why is this subset special? How does it help determine the range of T?

Thanks.

 

[latentcorpse]

$dim(P^3)=dim(\mahtbb{R}^4)=4$

you have the basis elements of $P^3$.

The action of T on each of these basis elements will let you know the basis elements of $\mathbb{R}^4$

i.e. $1,x,x^2,x^3$ are the basis elements of $P^3$
and $T(1),T(x),T(X^2),T(x^3)$ are the basis elemetns of $\mathbb{R}^4$

applying T to any element of $p(x) \in P^3$ will yield $T(p(x)) \in \mathbb{R}^4$ and $T(p(x))=aT(1)+bT(x)+cT(x^2)+dT(x^3)$ where $a,b,c,d \in \mathbb{Z}$

 

[Butelle]

thank you!