By A. R. Heesterman

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Extra resources for Handbook of linear algebra

Sample text

Then: r (AT )T = A r (A + B)T = AT + B T r (cA)T = cAT r (AB)T = B T AT 5. [Leo02, pp. 321–323] Let c ∈ C, let A and B be matrices over C, and assume the shapes allow the following sums and products to be calculated. Then: r (A∗ )∗ = A r (A + B)∗ = A∗ + B ∗ r (c A)∗ = c¯ A∗ r (AB)∗ = B ∗ A∗ 6. If A and B are n × n and upper (lower) triangular, then AB is upper (lower) triangular. ⎡ Examples: ⎤ 7 1 2 3 1 2 3 −4 1. Let A = and b = ⎣ 8⎦. By definition, Ab = 7 +8 −9 = . Hand 4 5 6 4 5 6 14 −9 1·7+2·8−3·9 −4 calculation of Ab can be done more quickly using Fact 2: Ab = = .

The following are equivalent: r S is a basis for V . r S spans V . r S is linearly independent. 2-4 Handbook of Linear Algebra 10. If dim(V ) = n, then any subset of V containing more than n vectors is linearly dependent. 11. If dim(V ) = n, then any subset of V containing fewer than n vectors does not span V . 12. 4] If B = {b1 , . . , b p } is a basis for a vector space V , then each x ∈ V can be expressed as a unique linear combination of the vectors in B. That is, for each x ∈ V there is a unique set of scalars c 1 , c 2 , .

1 2-1 2-3 2-4 2-6 2-9 2-10 2-12 2-12 Span and Linear Independence Let V be a vector space over a field F . Definitions: A linear combination of the vectors v1 , v2 , . . , vk ∈ V is a sum of scalar multiples of these vectors; that is, c 1 v1 + c 2 v2 + · · · + c k vk , for some scalar coefficients c 1 , c 2 , . . , c k ∈ F . If S is a set of vectors in V , a linear combination of vectors in S is a vector of the form c 1 v1 + c 2 v2 + · · · + c k vk with k ∈ N, vi ∈ S, c i ∈ F . Note that S may be finite or infinite, but a linear combination is, by definition, a finite sum.