Find the matrix A' = [LA]. Also find an invertible matrix Q such that A' = Q-¹AQ. (a) Suppose W₁ and W₂ are subspaces in a vector space V and that V = W₁ + W₂. Prove that V = W₁ W₂ if and only if any vector v € V can be represented uniquely as v = v₁ + v2 where V₁ € W₁, v₂ € W₂. (b) Suppose W₁ and W₂ are subspaces in a vector space V, and such that V =W₁ W₂. If B₁ is a basis of W₁ and 32 is a basis of W2 prove that 31 U 32 is a basis of V.
Find the matrix A' = [LA]. Also find an invertible matrix Q such that A' = Q-¹AQ. (a) Suppose W₁ and W₂ are subspaces in a vector space V and that V = W₁ + W₂. Prove that V = W₁ W₂ if and only if any vector v € V can be represented uniquely as v = v₁ + v2 where V₁ € W₁, v₂ € W₂. (b) Suppose W₁ and W₂ are subspaces in a vector space V, and such that V =W₁ W₂. If B₁ is a basis of W₁ and 32 is a basis of W2 prove that 31 U 32 is a basis of V.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Transcription for Educational Website**
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**Find the matrix \( A' = [L_A]_\beta^\beta \). Also find an invertible matrix \( Q \) such that \( A' = Q^{-1}AQ \).**
(a) Suppose \( W_1 \) and \( W_2 \) are subspaces in a vector space \( V \) and that \( V = W_1 + W_2 \). Prove that \( V = W_1 \oplus W_2 \) if and only if any vector \( v \in V \) can be represented uniquely as \( v = v_1 + v_2 \) where \( v_1 \in W_1, v_2 \in W_2 \).
(b) Suppose \( W_1 \) and \( W_2 \) are subspaces in a vector space \( V \), and such that \( V = W_1 \oplus W_2 \). If \( \beta_1 \) is a basis of \( W_1 \) and \( \beta_2 \) is a basis of \( W_2 \), prove that \( \beta_1 \cup \beta_2 \) is a basis of \( V \).
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Transcribed Image Text:**Transcription for Educational Website**
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**Find the matrix \( A' = [L_A]_\beta^\beta \). Also find an invertible matrix \( Q \) such that \( A' = Q^{-1}AQ \).**
(a) Suppose \( W_1 \) and \( W_2 \) are subspaces in a vector space \( V \) and that \( V = W_1 + W_2 \). Prove that \( V = W_1 \oplus W_2 \) if and only if any vector \( v \in V \) can be represented uniquely as \( v = v_1 + v_2 \) where \( v_1 \in W_1, v_2 \in W_2 \).
(b) Suppose \( W_1 \) and \( W_2 \) are subspaces in a vector space \( V \), and such that \( V = W_1 \oplus W_2 \). If \( \beta_1 \) is a basis of \( W_1 \) and \( \beta_2 \) is a basis of \( W_2 \), prove that \( \beta_1 \cup \beta_2 \) is a basis of \( V \).
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