Integral linear combination Watch

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#1
Report Thread starter 8 years ago
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Let A,B,C be the standard orthornormal bases of Z^r, Z^s and Z^n respectively.

Let f be a normed bilinear map, f:Z^r x Z^s -> Z^n then for a in A and b in B, f(a,b) is also a unit vector and an integral linear combination of the basis vectors in C.

Can someone explain what the bold means?
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DFranklin
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#2
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Don't know; the obvious guess would be that it can be written in the form \sum \lambda_i e_i where e_i are the basis vectors and \lambda_i \in \mathbb{Z}. But since this is trivially true for the standard basis vectors, it doesn't feel it can be right.
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username167718
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#3
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I agree with DFranklin's interpretation of it...
Since the range of f is Z^n, you can trivially write it as a linear combination of the basis vectors, where you have an integer multiple of each vector.
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#4
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Thank you both - that's probably what is meant. After looking at the notes again, it's clear that the writer is just highlighting the fact that there are integers involved.
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#5
Report Thread starter 8 years ago
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Using the same notation from my first post:



Can someone explain the final equality? The previous equality is due to the fact that f is bilinear but I don't see how the norm of the sum is evaluated.
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