Question

Consider that $T: U \to V$ and $S: V \to W$ are linear transformations and $B$, $C$, and $D$ are bases for $U$, $V$, and $W$, respectively. Compute $[S \circ T]_{D \leftarrow B}$ in two ways. $T: P_1 \to \mathbb{R}^2$ defined by $T(p(x)) = \begin{bmatrix} p(0) \ p(1) \end{bmatrix}$, $S: \mathbb{R}^2 \to \mathbb{R}^2$ defined by $S\begin{bmatrix} a \ b \end{bmatrix} = \begin{bmatrix} a - 2b \ 2a - b \end{bmatrix}$, $B = \{1, x\}$, $C = D = \{e_1, e_2\}$ (a) by finding $S \circ T$ directly and then computing its matrix $[S \circ T]_{D \leftarrow B} = $ (b) by finding the matrices of $S$ and $T$ separately and using the theorem below. Let $U$, $V$, and $W$ be finite-dimensional vector spaces with bases $B$, $C$, and $D$, respectively. Let $T: U \to V$ and $S: V \to W$ be linear transformations. Then $[S \circ T]_{D \leftarrow B} = [S]_{D \leftarrow C}[T]_{C \leftarrow B}$ $[S \circ T]_{D \leftarrow B} = $

          Consider that $T: U \to V$ and $S: V \to W$ are linear transformations and $B$, $C$, and $D$ are bases for $U$, $V$, and $W$, respectively.
Compute $[S \circ T]_{D \leftarrow B}$ in two ways.
$T: P_1 \to \mathbb{R}^2$ defined by $T(p(x)) = \begin{bmatrix} p(0) \ p(1) \end{bmatrix}$, $S: \mathbb{R}^2 \to \mathbb{R}^2$ defined by $S\begin{bmatrix} a \ b \end{bmatrix} = \begin{bmatrix} a - 2b \ 2a - b \end{bmatrix}$, $B = \{1, x\}$,
$C = D = \{e_1, e_2\}$
(a) by finding $S \circ T$ directly and then computing its matrix
$[S \circ T]_{D \leftarrow B} = 
$
(b) by finding the matrices of $S$ and $T$ separately and using the theorem below.
Let $U$, $V$, and $W$ be finite-dimensional vector spaces with bases $B$, $C$, and $D$, respectively. Let $T: U \to V$ and $S: V \to W$ be linear transformations. Then
$[S \circ T]_{D \leftarrow B} = [S]_{D \leftarrow C}[T]_{C \leftarrow B}$
$[S \circ T]_{D \leftarrow B} = $

$Consider that T: U → V and S: V → W are linear transformations and B, C, and D are bases for U, V, and W, respectively. Compute [S ∘ T]D ← B in two ways. T: P1 →ℝ^2 defined by T(p(x)) = < b m a t r i x >, S: ℝ^2 →ℝ^2 defined by S < b m a t r i x > = < b m a t r i x >, B = {1, x}, C = D = {e1, e2} (a) by finding S ∘ T directly and then computing its matrix [S ∘ T]D ← B = (b) by finding the matrices of S and T separately and using the theorem below. Let U, V, and W be finite-dimensional vector spaces with bases B, C, and D, respectively. Let T: U → V and S: V → W be linear transformations. Then [S ∘ T]D ← B = [S]D ← C[T]C ← B [S ∘ T]D ← B =$

Added by Laura L.

Question

Please give Ace some feedback