Question

3. Consider the vector space $V = P^{\leq 2} = \{ax^2 + bx + c : a, b, c \in \mathbb{R}\}$ of quadratic polynomials. Let $\mathcal{B}$ be the ordered basis $(1, x + 2, x^2 + 2x + 3)$. Let $\mathcal{C}$ be the ordered basis $(x^2, (x + 1)^2, (x - 1)^2)$. (a) Find the change-of-basis matrices $M_{\mathcal{C} \leftarrow \mathcal{B}}$ and $M_{\mathcal{B} \leftarrow \mathcal{C}}$. (b) Let $T : V \to V$ be the derivative operation. Find the matrix representations of $T$ with respect to $\mathcal{B}$ and $\mathcal{C}$.

          3. Consider the vector space $V = P^{\leq 2} = \{ax^2 + bx + c : a, b, c \in \mathbb{R}\}$ of quadratic polynomials.

Let $\mathcal{B}$ be the ordered basis $(1, x + 2, x^2 + 2x + 3)$. Let $\mathcal{C}$ be the ordered basis $(x^2, (x + 1)^2, (x - 1)^2)$.

(a) Find the change-of-basis matrices $M_{\mathcal{C} \leftarrow \mathcal{B}}$ and $M_{\mathcal{B} \leftarrow \mathcal{C}}$.
(b) Let $T : V \to V$ be the derivative operation. Find the matrix representations of $T$ with respect to $\mathcal{B}$ and $\mathcal{C}$.

$3. Consider the vector space V = P^≤ 2 = {ax^2 + bx + c : a, b, c ∈ℝ} of quadratic polynomials. Let ℬ be the ordered basis (1, x + 2, x^2 + 2x + 3). Let 𝒞 be the ordered basis (x^2, (x + 1)^2, (x - 1)^2). (a) Find the change-of-basis matrices M𝒞←ℬ and Mℬ←𝒞. (b) Let T : V → V be the derivative operation. Find the matrix representations of T with respect to ℬ and 𝒞.$

Added by Nancy H.

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