Math Problem Statement
Let T : R3 → R3 be the linear transformation defined by T (x, y, z) = (x + y, y + z, x + z). Let E = {(1, 0, 0), (0, 1, 0), (0, 0, 1)} be the standard basis for R3, and let B = {(1, 1, 0), (0, 1, 1), (1, 0, 1)} be another basis for R3.
- Find the matrix [T ]E of T relative to the basis E. That is [T ]E = T 1 0 0 E T 0 1 0 E T 0 0 1 E
- Find the change-of-basis matrix T E B .
- Find the matrix [T ]B of T relative to the basis B. That is [T ]B = T 1 1 0 B T 0 1 1 B T 1 0 1
Solution
Solution
1. Find the matrix of relative to the standard basis .
To find , we apply to each basis vector in and express the results in terms of the standard basis:
- .
- .
- .
Thus, is:
\begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{bmatrix}.$$ --- #### 2. Find the change-of-basis matrix $$T_E^B$$. The change-of-basis matrix $$T_E^B$$ converts coordinates from the $$B$$-basis to the $$E$$-basis. To find it, write each vector in $$B = \{(1, 1, 0), (0, 1, 1), (1, 0, 1)\}$$ as a linear combination of the standard basis $$E$$. Since $$B$$ is already expressed in $$E$$, the columns of $$T_E^B$$ are just the vectors in $$B$$ expressed in terms of $$E$$: $$T_E^B = \begin{bmatrix} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix}.$$ --- #### 3. Find the matrix $$[T]_B$$ of $$T$$ relative to the basis $$B$$. To compute $$[T]_B$$, we use the relation: \[ [T]_B = (T_E^B)^{-1} \cdot [T]_E \cdot T_E^B. \] First, compute $$(T_E^B)^{-1}$$. Using standard inversion techniques for a $$3 \times 3$$ matrix, we find: $$(T_E^B)^{-1} = \begin{bmatrix} 1 & -1 & -1 \\ -1 & 1 & 0 \\ -1 & 0 & 1 \end{bmatrix}.$$ Next, compute the product $$[T]_E \cdot T_E^B$$: $$[T]_E \cdot T_E^B = \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix} = \begin{bmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{bmatrix}.$$ Finally, compute: \[ [T]_B = (T_E^B)^{-1} \cdot ([T]_E \cdot T_E^B): \] $$[T]_B = \begin{bmatrix} 1 & -1 & -1 \\ -1 & 1 & 0 \\ -1 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{bmatrix} = \begin{bmatrix} 1 & -1 & 0 \\ 0 & 1 & -1 \\ -1 & 0 & 1 \end{bmatrix}.$$ --- ### Final Answers 1. $$[T]_E = \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{bmatrix}$$. 2. $$T_E^B = \begin{bmatrix} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix}$$. 3. $$[T]_B = \begin{bmatrix} 1 & -1 & 0 \\ 0 & 1 & -1 \\ -1 & 0 & 1 \end{bmatrix}$$. --- Would you like more detailed explanations for any part? Here are some related questions to explore: 1. What is the geometric interpretation of $$[T]_E$$ and $$[T]_B$$? 2. How is the change-of-basis matrix $$T_E^B$$ constructed? 3. Why is matrix multiplication used to find $$[T]_B$$ from $$[T]_E$$? 4. How would you verify that $$[T]_B$$ is correct? 5. What happens if the bases $$E$$ and $$B$$ are not linearly independent? **Tip:** Always ensure that your change-of-basis matrix is invertible before proceeding with computations, as it ensures the new basis is valid!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Transformations
Matrix Representation
Change of Basis
Basis Vectors
Formulas
[T]_E = [T(v1)_E, T(v2)_E, T(v3)_E]
Change-of-basis matrix: T_E^B = [B in terms of E]
[T]_B = (T_E^B)^(-1) · [T]_E · T_E^B
Theorems
Linear Independence and Basis
Matrix Multiplication for Transformation Composition
Suitable Grade Level
Undergraduate Mathematics (Linear Algebra)
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