How to find the basis for $\mathrm{Ker}(T)$ of the following linear transformation?

Question

$\newcommand{\Ker}{\mathrm{Ker}}$Find a basis for the vector space $V$ so that the first $\dim(\Ker(T))$ vectors are a basis for $\Ker(T)$, $T:V\rightarrow W $ a linear transformation.

1. $T:R^4 \rightarrow P_2(R)$ defined by $T(a_1, ..., a_4)=(a_1+a_2)+(a_2+a_3)x+(a_3+a_4)x^2$

2. $T:P_n(R) \rightarrow P_n(R)$, which is given by differentiation.

In terms of 1), does the matrix look something like $\begin{bmatrix}a_1+a_2&0&0\\0&a_2+a_3&0\\0&0&a_3+a_4\end{bmatrix}$? The answer says that $\{(1,-1,1,-1), e_1, e_2, e_3\}$ is the basis, but why?

For 2), the answer says $\{1, x, ..., x^n\}$ is the basis, but I still cannot write out the matrix.

Answer 1

Hints:

1. For your first problem
   • Do you see how to get each of $1$, $x$, and $x^2$ in the image (using $e_1$, $e_2$, and $e_3$)?
   • Do you see that this implies that the image is all of $P_2(R)$ so the kernel is one dimensional?
   • Are you able to solve $a_1 + a_2 = 0, a_2 + a_3 = 0, a_3 + a_4 = 0$? Do you see that the first given vector solves this system?
2. For your second problem
   • Are you able to take the derivative of $x$? What is it? Where does this tell you is a $1$ in your matrix?
   • Are you able to take the derivative of $x^2$? What is it? Where does this tell you is a $2$ in your matrix?
   • Are you able to take the derivative of $x^3$? What is it? Where does this tell you is a $3$ in your matrix?
   • $\vdots$

Answer 2

for (1), choose the standard basis $\{(1,0,0,0)^\top, (0,1,0,0)^\top, \cdots\}$ in $R^4$ and $\{1, x, x^2\}$ for $P_2(R).$ with respect to these basis $T$ has the matrix representation $\pmatrix{1&1&0&0\\0&1&1&0\\0&0&1&1}.$