Question

The space $P_2$ is the vector space of polynomials of degree 2 or less. Recall that a polynomial such as \begin{equation} p(x) = 7 + 6x + 5x^2 \text{ would be the vector } \begin{bmatrix} 7\\6\\5 \end{bmatrix} \text{ in } \mathbb{R}^3.\end{equation} The standard basis polynomials for this space are $\{1, x, x^2\}$. The function $F$, defined by $F(p(x)) = \begin{bmatrix} p(-4) \\ p'(-4) \end{bmatrix}$, is a linear transformation from $P_2$ to $\mathbb{R}^2$. It evaluates the polynomial and its first derivative, both at the input $x = -4$, and uses those values as the components of a vector. Find $F(4 - 2x - x^2)$. Answer: $\begin{bmatrix} -4\\6 \end{bmatrix}$ What should be the size of the matrix representing the function $F$? Answer: 2 rows and 3 columns. Write the matrix for this linear transformation according to the standard basis polynomials. [Hint: Find where the standard basis polynomials go under this transformation.] $\begin{bmatrix} 1 & -4 & 16\\0 & 1 & -8 \end{bmatrix}$ Now draw three different non-zero polynomials that are in the null space of this transformation. Null space?

          The space $P_2$ is the vector space of polynomials of degree 2 or less. Recall that a polynomial such as
\begin{equation*}
p(x) = 7 + 6x + 5x^2 \text{ would be the vector } \begin{bmatrix} 7\\6\\5 \end{bmatrix} \text{ in } \mathbb{R}^3.\end{equation*}
The standard basis polynomials for this space are
\(\{1, x, x^2\}\).
The function $F$, defined by $F(p(x)) = \begin{bmatrix} p(-4) \\ p'(-4) \end{bmatrix}$, is a linear transformation from $P_2$ to $\mathbb{R}^2$. It evaluates
the polynomial and its first derivative, both at the input $x = -4$, and uses those values as the components
of a vector.
Find $F(4 - 2x - x^2)$.
Answer:
$\begin{bmatrix} -4\\6 \end{bmatrix}$
What should be the size of the matrix representing the function $F$?
Answer: 2 rows and 3 columns.
Write the matrix for this linear transformation according to the standard basis polynomials. [Hint: Find
where the standard basis polynomials go under this transformation.]
$\begin{bmatrix} 1 & -4 & 16\\0 & 1 & -8 \end{bmatrix}$
Now draw three different non-zero polynomials that are in the null space of this transformation.
Null space?

$The space P2 is the vector space of polynomials of degree 2 or less. Recall that a polynomial such as p(x) = 7 + 6x + 5x^2 would be the vector < b m a t r i x > in ℝ^3. The standard basis polynomials for this space are {1, x, x^2}. The function F, defined by F(p(x)) = < b m a t r i x >, is a linear transformation from P2 to ℝ^2. It evaluates the polynomial and its first derivative, both at the input x = -4, and uses those values as the components of a vector. Find F(4 - 2x - x^2). Answer: < b m a t r i x > What should be the size of the matrix representing the function F? Answer: 2 rows and 3 columns. Write the matrix for this linear transformation according to the standard basis polynomials. [Hint: Find where the standard basis polynomials go under this transformation.] < b m a t r i x > Now draw three different non-zero polynomials that are in the null space of this transformation. Null space?$

Added by -Ngeles P.

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