# Give a basis for $\mathrm{Ker}(T)$ and $\mathrm{Im}(T)$ of the linear transformations

I was given the following problem:

Let $T:\mathbb {R^3} \rightarrow \mathbb {P_2(\mathbb {R})}$ and $G:\mathbb {P_2(\mathbb {R})} \rightarrow \mathbb {R^3}$ Linear transformations such that:

$[ T]_{B,C} = \begin{pmatrix} 1 & 2 & -1 & \\ 1 & 0 & -1 & \\ 0 & 1 & 1 & \\ \end{pmatrix}$

$[ G]_{C,B} = \begin{pmatrix} 1 & 2 & 2 & \\ 1 & -1 & 0 & \\ -1 & 1 & 0 & \\ \end{pmatrix}$

where $B=\{(1,1,0),(0,1,0),(0,0,1)\}$ and $C=\{1,1+x,1+x^2\}$

a)Give the basis for $KerT$ and $ImT$

b)Give basis for $Ker(G\circ T)$ and $ImT(G\circ T)$

c)give the matrix of $H=3(T\circ G)+I$ with respect to the basis $\{1,x,x^2\}$

Then my doubt is how to find the kernel and image of the linear transformation. The doubt is basically in the part a) of the problem. Having got that, I guess I can solve the other parts.

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Looks like homework.. – Your Ad Here Oct 22 '12 at 19:28

a) $T$ is invertible. Therefore Kernel is 0 Image the whole thing.
b) Kernel of $G$ are multiples of $(2,2,-3)$.