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1. Show that WW10 W2 is a vector subspace of V, but that W1 U W2 need not be a vector subspace in general. 2. Show that

1. Show that WW10 W2 is a vector subspace of V, but that W1 U W2 need not be a vector subspace in general. 2. Show that

Advanced Engineering Mathematics
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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Let W1 and W½ be two subspaces of a finite dimensional vector space V over a field F. 1. Show that Win W2 is a vector subspace of V, but that Wi U W2 need not be a vector subspace in general. 2. Show that dim(W1) + dim(W2) – dim(W1 n W2) = dim(W1 + W2), where W1 + W2 just denotes the span of W1 U W2 in V. (Hint: Apply the rank nullity theorem to the natural map W1 O W2 → V.) 3. (Cultural comment, for extra credit.) The formula of part 2 is reminiscent of the simple combinatorial formula #S1 + #S2 – #(S1 n S2) = #(SI U S2). How might you deduce t from the formula of part 2? (f course this is a bit perverse since (3) is arguably easier than (2), but it illustrates the important way in which linear algebra resonates with combinatorics.)
Transcribed Image Text:Let W1 and W½ be two subspaces of a finite dimensional vector space V over a field F. 1. Show that Win W2 is a vector subspace of V, but that Wi U W2 need not be a vector subspace in general. 2. Show that dim(W1) + dim(W2) – dim(W1 n W2) = dim(W1 + W2), where W1 + W2 just denotes the span of W1 U W2 in V. (Hint: Apply the rank nullity theorem to the natural map W1 O W2 → V.) 3. (Cultural comment, for extra credit.) The formula of part 2 is reminiscent of the simple combinatorial formula #S1 + #S2 – #(S1 n S2) = #(SI U S2). How might you deduce t from the formula of part 2? (f course this is a bit perverse since (3) is arguably easier than (2), but it illustrates the important way in which linear algebra resonates with combinatorics.)
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