Let V be an F-vector space and A, B, C are subspaces of V. (a) Show that the equation An (B+C) = (ANB) + (ANC) is not necessarily true by an example. (b) Prove that An (B+ (ANC)) (ANB) + (Anc). =
Let V be an F-vector space and A, B, C are subspaces of V. (a) Show that the equation An (B+C) = (ANB) + (ANC) is not necessarily true by an example. (b) Prove that An (B+ (ANC)) (ANB) + (Anc). =
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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
Transcribed Image Text:**Exercise: Vector Spaces and Subspaces**
Let \( \mathcal{V} \) be an \( \mathbb{F} \)-vector space and \( \mathcal{A}, \mathcal{B}, \mathcal{C} \) are subspaces of \( \mathcal{V} \).
(a) Show that the equation \( \mathcal{A} \cap (\mathcal{B} + \mathcal{C}) = (\mathcal{A} \cap \mathcal{B}) + (\mathcal{A} \cap \mathcal{C}) \) is not necessarily true by an example.
(b) Prove that \( \mathcal{A} \cap (\mathcal{B} + (\mathcal{A} \cap \mathcal{C})) = (\mathcal{A} \cap \mathcal{B}) + (\mathcal{A} \cap \mathcal{C}) \).
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