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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: λ (( a ij )) = (( λa 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 λ ∈ K , A ∈ W then λA ∈ 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 ( λA ) T = λA T = λA so λA ∈ 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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This note was uploaded on 05/31/2011 for the course MATH 224 taught by Professor Y.burda during the Spring '11 term at University of Toronto.
 Spring '11
 Y.Burda
 Algebra, Addition, Multiplication, Matrices, Scalar, Vector Space

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