LinearAlgebraLecture2

LinearAlgebraLecture2 - Another example of a vector space:...

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Unformatted text preview: Another example of a vector space: K m n- the space of m n matrices with entries from the field 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: (( a ij )) = (( a ij )). Problem : Prove that the set of antisymmetric matrices { M K 3 3 | M T =- M } is a subspace of K 3 3 and find 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 T + B T =- A- B =- ( A + B ), so A + B W 2. If A T =- A , then ( A ) T = A T =- A so A W To find the dimension, we will find 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)....
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LinearAlgebraLecture2 - Another example of a vector space:...

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