Questions: Finding Transition and Coordinate Matrices In Exercises 37-40, (a) find the transition matrix from B to B', (b) find the transition matrix from B' to B, (c) verify that the two transition matrices are inverses of each other, and (d) find the coordinate matrix [x]B, given the coordinate matrix [x]B'. 37. B=(1,3),(-2,-2), B'=(-12,0),(-4,4), [x]B'= -1 3

Solution Steps

To solve this problem, we need to perform the following steps: (a) To find the transition matrix from $B$ to $B'$, express each vector in $B$ as a linear combination of the vectors in $B'$. The coefficients of these combinations form the columns of the transition matrix. (b) To find the transition matrix from $B'$ to $B$, express each vector in $B'$ as a linear combination of the vectors in $B$. The coefficients of these combinations form the columns of the transition matrix. (c) Verify that the two transition matrices are inverses by multiplying them and checking if the result is the identity matrix. (d) To find the coordinate matrix $[\mathbf{x}]_B$, use the transition matrix from $B'$ to $B$ and multiply it by $[\mathbf{x}]_{B'}$.

We have the basis vectors: $$B = \{(1, 3), (-2, -2)\}, \quad B' = \{(-12, 0), (-4, 4)\}$$

The transition matrix from $B$ to $B'$ is given by: $$T_{B \to B'} = \begin{bmatrix} -0.0833 & -0.25 \\ -0.5833 & -0.75 \end{bmatrix}$$

The transition matrix from $B'$ to $B$ is: $$T_{B' \to B} = \begin{bmatrix} 9 & -3 \\ -7 & 1 \end{bmatrix}$$

To verify that the two transition matrices are inverses, we check: $$T_{B \to B'} \cdot T_{B' \to B} = I$$ where $I$ is the identity matrix: $$I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$ The result confirms that $T_{B \to B'}$ and $T_{B' \to B}$ are indeed inverses, as $are\_inverses = \text{True}$.

Given the coordinate matrix $[\mathbf{x}]_{B'} = \begin{bmatrix} -1 \\ 3 \end{bmatrix}$, we find: $$[\mathbf{x}]_B = T_{B' \to B} \cdot [\mathbf{x}]_{B'} = \begin{bmatrix} -18 \\ 10 \end{bmatrix}$$

Final Answer