The given problem defines a linear transformation $T: P_2 \to \mathbb{R}^2$ by the following mapping:

$T(p) = \begin{bmatrix} p(0) \\ p(0) \end{bmatrix}$

Here, $P_2$ is the space of polynomials of degree at most 2, meaning any polynomial in $P_2$ has the form:

$p(x) = ax^2 + bx + c$

1. Finding the Kernel of $T$

The kernel of $T$, $\text{ker}(T)$, consists of all polynomials $p(x) \in P_2$ such that:

$T(p) = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$

This implies:

$p(0) = 0$

Substituting $x = 0$ into $p(x) = ax^2 + bx + c$, we get:

$p(0) = c$

So, for $T(p) = 0$, we must have $c = 0$. Thus, the polynomials in the kernel of $T$ are those of the form:

$p(x) = ax^2 + bx$

The kernel is spanned by the polynomials:

$p_1(x) = x \quad \text{and} \quad p_2(x) = x^2$

Thus, the kernel of $T$ is spanned by $\{x, x^2\}$.

2. Describing the Range of $T$

The range of $T$, $\text{range}(T)$, consists of all vectors of the form $\begin{bmatrix} p(0) \\ p(0) \end{bmatrix}$. Since the output vector depends only on the constant term $c$ of the polynomial $p(x)$, we have:

$T(p) = \begin{bmatrix} c \\ c \end{bmatrix}$

Thus, the range of $T$ is spanned by the vector $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$. Therefore:

$\text{range}(T) = \text{span} \left( \begin{bmatrix} 1 \\ 1 \end{bmatrix} \right)$

Conclusion:

• The kernel of $T$ is spanned by $\{x, x^2\}$.
• The range of $T$ is spanned by $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$.

Would you like further details on any part of this problem?

Here are five related questions to deepen your understanding:

1. How does the rank-nullity theorem apply to this transformation $T$?
2. What is the dimension of the kernel and range of $T$?
3. How would you describe the geometric interpretation of the kernel and range?
4. Can the transformation $T$ be represented by a matrix, and if so, what would it look like?
5. How does changing the definition of $T$ (e.g., modifying the output components) affect the kernel and range?

Tip: Always check the rank of a linear transformation to understand how many independent vectors map to nonzero outputs.