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Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.7: Applications

Problem 74EQ

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Question

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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