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bmanbs2
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Homework Statement
Consider integer sequence [tex]n_{1},...,n_{r}[/tex] and matrices [tex]A_{1},...,A_{n-1}[/tex]. Assume [tex]im\left(A_{i}\right) = ker\left(A_{i+1}\right)[/tex]
Using the rank-nullity theorem, show that [tex]\sum^{n}_{i=1}\left(-1\right)^{i}d_{i} = 0[/tex]
Homework Equations
The rank-nullity theorem states that if [tex]v[/tex] and [tex]w[/tex] are vector spaces and [tex]A[/tex] is the linear map A: v -> w, then
[tex]dim\left(im\left(A\right)\right) + dim\left(ker\left(A\right)\right) = dim\left(v\right)[/tex]
The Attempt at a Solution
I know that the relation [tex]im\left(A_{i}\right) = ker\left(A_{i+1}\right)[/tex] means that [tex]\sum^{n}_{i even}d_{i}[/tex] = [tex]\sum^{n}_{i odd}d_{i}[/tex], but I don't know how to get there.