Examples, Subspaces

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Martin, Burda

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Another example of a vector space: K mn - the space of m n matrices with entries from the eld K. This is a vector space over K with the addition being the usual matrix addition and scalar multiplication being the entry- wise multiplication by scalars: ((ij)) = ((aij). Problem: Prove that the set of antisymmetric matrices {M K M T = 33 M} is a subspace of K 33 and nd its dimension. Solution: To show W is a subspace we should verify these two conditions: 1. If A,B W then A + B W 2. If K,A W then A W. Indeed: 1. If A T = A and B T = B, then (A + B) T = A + B T = A B = (A + B), so A + B W T T T 2. If A = A, then (A) = A = A so A W To nd the dimension, we will nd a spanning set and show that it is linearly independent (or eliminate the dependent elements from it, if it turns out to be linearly dependent). a b c a d g a b c According to the denition of W, W = { d e f b e h = d e f }. g h k c f k g h k 0 b c 0 1 0 In other words W = { b 0 f } = {bE 1cE +f2 }, w3ere E = 1 1 0 0
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