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. Show that dim(w,) + dim(W,)– dim(w, n w,) = dim(w, + w, - 1 C

. Show that dim(w,) + dim(W,)– dim(w, n w,) = dim(w, + w, - 1 C

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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2. Show that dim(w,) + dim(w,) – dim(w, n w,) = dim(W, + W,) %3D 1 1 FW,uW2 → V.) where W, + W, just denotes the span of W, U W, in V. (Hint: Apply the rank nullity theorem to the natural map W, O W, 3. The formula of part 2 is reminiscent of the simple combinatorial formula #S, + #S. 1 #(s, n s.) = #(s, u s.) 2 1 How might you deduce it from the formula of part 2? (of 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:2. Show that dim(w,) + dim(w,) – dim(w, n w,) = dim(W, + W,) %3D 1 1 FW,uW2 → V.) where W, + W, just denotes the span of W, U W, in V. (Hint: Apply the rank nullity theorem to the natural map W, O W, 3. The formula of part 2 is reminiscent of the simple combinatorial formula #S, + #S. 1 #(s, n s.) = #(s, u s.) 2 1 How might you deduce it from the formula of part 2? (of 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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