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Let Wi and W2 be two subspaces of a finite dimensional vector space V over a field F. 1. Show that W1 n W, is a vector subspace of V, but that W¡ 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 Wi O W2 → V .) 3. (Cultural comment, for extra credit.) The formula of part 2 is reminiscent of the simple combinatorial formula #S1 + #S2 – #(Si n S2) = #(S1 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.)

Let Wi and W2 be two subspaces of a finite dimensional vector space V over a field F. 1. Show that W1 n W, is a vector subspace of V, but that W¡ 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 Wi O W2 → V .) 3. (Cultural comment, for extra credit.) The formula of part 2 is reminiscent of the simple combinatorial formula #S1 + #S2 – #(Si n S2) = #(S1 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.)

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 W2 be two subspaces of a finite dimensional vector space V over a field F. 1. Show that Wi n Wz 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(Wi + W2), where W1 + W2 just denotes the span of W1 U W2 in V. (Hint: Apply the rank nullity theorem to the natural map Wi 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) = #(S1 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 W2 be two subspaces of a finite dimensional vector space V over a field F. 1. Show that Wi n Wz 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(Wi + W2), where W1 + W2 just denotes the span of W1 U W2 in V. (Hint: Apply the rank nullity theorem to the natural map Wi 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) = #(S1 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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