Question

3. Let V be the span of the functions $\cos(2x)$ and $\sin(2x)$. The sets $\alpha = \{\cos(2x), \sin(2x)\}$ $\beta = \{\cos(2x) + \sin(2x), \cos(2x) - \sin(2x)\}$ are bases for V. Let $L = D^2 + 1$. (a) (6 points) Compute the change of basis matrix $P = [I]_\beta^\alpha$ and its inverse. (b) (3 points) Compute the matrix of L with respect to $\alpha$. (c) (3 points) Compute the matrix of L with respect to $\beta$. (d) (2 points) Find $[3\cos(2x) + 4\sin(2x)]_\beta$. (e) (1 point (bonus)) Describe geometrically what L does to V.

          3. Let V be the span of the functions $\cos(2x)$ and $\sin(2x)$. The sets
$\alpha = \{\cos(2x), \sin(2x)\}$
$\beta = \{\cos(2x) + \sin(2x), \cos(2x) - \sin(2x)\}$
are bases for V. Let $L = D^2 + 1$.
(a) (6 points) Compute the change of basis matrix $P = [I]_\beta^\alpha$ and its inverse.
(b) (3 points) Compute the matrix of L with respect to $\alpha$.
(c) (3 points) Compute the matrix of L with respect to $\beta$.
(d) (2 points) Find $[3\cos(2x) + 4\sin(2x)]_\beta$.
(e) (1 point (bonus)) Describe geometrically what L does to V.

$3. Let V be the span of the functions cos(2x) and sin(2x). The sets α = {cos(2x), sin(2x)} β = {cos(2x) + sin(2x), cos(2x) - sin(2x)} are bases for V. Let L = D^2 + 1. (a) (6 points) Compute the change of basis matrix P = [I]β^α and its inverse. (b) (3 points) Compute the matrix of L with respect to α. (c) (3 points) Compute the matrix of L with respect to β. (d) (2 points) Find [3cos(2x) + 4sin(2x)]β. (e) (1 point (bonus)) Describe geometrically what L does to V.$

Added by Tony L.

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