Math Problem Statement
Find the matrix of the linear operator A in the basis {e1, e2, e3} given its matrix in the basis {a1, a2, a3} and the decomposition of a1, a2, a3 in terms of e1, e2, e3.
Solution
To solve this problem, we need to find the matrix of the linear operator in the basis , given the matrix in the basis and the decomposition of in terms of .
Here are the steps:
Step 1: Representing in the basis
The problem provides:
\mathbf{a}_2 = \mathbf{e}_1 - 2\mathbf{e}_3, \quad \mathbf{a}_3 = \mathbf{e}_1 + 2\mathbf{e}_2$$ This gives the transformation matrix $$P$$ that maps the basis $$\{\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3\}$$ to $$\{\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3\}$$. Specifically, $$P$$ is formed as: $$P = \begin{bmatrix} 2 & 1 & 1 \\ 1 & 0 & 2 \\ 0 & -2 & 0 \end{bmatrix}$$ ### Step 2: Transition to the new basis To find the matrix $$B$$ of the linear operator in the basis $$\{\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3\}$$, we use the formula: $$B = P^{-1}AP$$ where $$A = \begin{bmatrix} 2 & 1 & 1 \\ 3 & 1 & 0 \\ -2 & -1 & 2 \end{bmatrix}$$. ### Step 3: Compute $$P^{-1}$$ Using standard techniques, compute the inverse of $$P$$. Let me calculate the inverse and the resulting matrix $$B$$. numpy as np # Given matrices A = np.array([[2, 1, 1], [3, 1, 0], [-2, -1, 2]]) P = np.array([[2, 1, 1], [1, 0, 2], [0, -2, 0]]) # Compute the inverse of P P_inv = np.linalg.inv(P) # Compute B = P^(-1) * A * P B = P_inv @ A @ P BThe matrix $$B$$ of the linear operator in the basis $$\{\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3\}$$ is: $$B = \begin{bmatrix} -0.67 & -3 & -0.33 \\ 2.5 & 3 & 2 \\ 3.83 & 3 & 2.67 \end{bmatrix}$$ Let me know if you want further details about the calculation or any clarifications. Here are five follow-up questions to deepen your understanding: 1. How is the transformation matrix $$P$$ constructed in general for any change of basis? 2. Why is the formula $$B = P^{-1}AP$$ used to transition between bases? 3. What properties does the matrix $$A$$ preserve in the new basis? 4. How can you verify the correctness of the resulting matrix $$B$$? 5. What are the practical applications of changing the basis in linear algebra? **Tip:** When transitioning between bases, always ensure the transformation matrix $$P$$ is invertible.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear algebra
Change of basis
Matrix transformations
Formulas
Matrix transformation: B = P^(-1)AP
Matrix inverse formula
Theorems
Change of basis theorem
Invertibility of linear transformations
Suitable Grade Level
Undergraduate (Linear Algebra)