I am working on this practice problem, and I was wondering if I could get some help.
I have a $T$:$\mathbb{R^{2×2}}\to \mathbf{P_{2}}$, that is, from 2×2 matrices to polynomials of degree at most 2. The transformation is given as following:
$$T\left(\begin{bmatrix}
a & b\\
c & d
\end{bmatrix}\right)
= a-c+2d+(b+2c-d)t+(a-c+3d)t^{2}.$$
To get the basis of kernel of $T$, I solved a system of equations needed to get the 'O' element in the $\mathbf{P_{2}}$ — $a-c+2d=0$, $b+2c-d=0$ and $a-c+3d=0$. As a result, I got the basis of the kernel equal to $$\begin{bmatrix}
1 & -2\\
1 & 0\\
\end{bmatrix}.$$
When it comes to image, if I understand correctly, I need to factor out all the variables separately, to see what is it that they span. So I got $a(1+t^{2})+b(t)+c(-t^{2}+2t-1)+d(3t^{2}-t+2)$. So would I be correct in saying that these three polynomials (without the coefficients $a$, $b$, $c$, and $d$) form the basis for the image $T$? Thank you!
Best Answer
Your answers are correct, but for the image part a) you need some additional reasoning and b) you can considerably simplify the result. a) These polynomials form a generating set for the image of $T$; to show that they form a basis, you have to check whether they're linearly independent (they are). b) You've correctly described the image space, but in quite a complicated way. What's the dimension of the space spanned by your basis of three polynomials? And what's the dimension of the entire codomain $\mathbf{P_2}$? What does that tell you about which space these elements actually span?
By the way, the basis element of the kernel should be written as a $2\times2$ matrix, not as a column vector, and the basis is the set containing that element, not the element itself.