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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?

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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.

Define a linear transformation T: P_2 → ℝ^2 by T(p) = [p(0) p(0)]. Find polynomials p1 and p2 in P_2 that span the kernel of T, and describe the range of T.

Explore how to find polynomials that span the kernel of the transformation T: P_2 → ℝ^2 and describe the range. Learn about the linear transformation T(p) = [p(0) p(0)] and its impact on polynomial space, suitable for undergraduate linear algebra students.

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