We are given a linear transformation $T : \mathbb{P}_2 \to \mathbb{R}^2$ defined by

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

where $\mathbb{P}_2$ is the space of polynomials of degree at most 2.

Step 1: Kernel of $T$

The kernel of $T$, denoted $\ker(T)$, is the set of polynomials $p(x) \in \mathbb{P}_2$ such that $T(p) = \mathbf{0}$. In other words, we want to find polynomials $p(x) \in \mathbb{P}_2$ such that:

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

This implies that:

$p(0) = 0.$

Thus, the polynomials in the kernel of $T$ must satisfy $p(0) = 0$. Any polynomial in $\mathbb{P}_2$ can be written as:

$p(x) = a_2 x^2 + a_1 x + a_0.$

For $p(0) = 0$, we require:

$a_0 = 0.$

Therefore, the polynomials in the kernel of $T$ are of the form:

$p(x) = a_2 x^2 + a_1 x.$

The set of polynomials that span the kernel are:

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

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

Step 2: Range of $T$

The range of $T$, denoted $\text{range}(T)$, is the set of all possible outputs of $T$. Since $T(p) = \begin{bmatrix} p(0) \\ p(0) \end{bmatrix}$, the output is always of the form:

$\begin{bmatrix} c \\ c \end{bmatrix}$

where $c = p(0)$. Therefore, the range of $T$ consists of all vectors in $\mathbb{R}^2$ where both components are equal. This is the subspace of $\mathbb{R}^2$ spanned by the vector:

$\begin{bmatrix} 1 \\ 1 \end{bmatrix}.$

Thus, the range of $T$ is the 1-dimensional subspace of $\mathbb{R}^2$ spanned by $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$.

Conclusion:

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

Would you like further details or have any other questions?

Follow-up questions:

1. How is the dimension of the kernel related to the dimension of the polynomial space $\mathbb{P}_2$?
2. What is the rank-nullity theorem, and how can it be applied to this linear transformation?
3. Can you describe the geometric interpretation of the range and kernel in this context?
4. What would change if $T(p)$ was defined differently, such as $T(p) = \begin{bmatrix} p(1) \\ p(-1) \end{bmatrix}$?
5. How would you verify that $\{x, x^2\}$ spans the kernel of $T$?

Tip:

To check if a set of polynomials spans a subspace, ensure that they are linearly independent and that any element of the subspace can be written as a linear combination of the set.