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2 Vector Spaces 35 (ai j ) + (bi j ) = (ai j + bi j ), α (ai j ) = (α ai j ). These definitions make Fm×n into a vector space. Given the matrix A = (ai j ), ˜ as the n × m matrix given by we define its transpose, which we denote by A, (a˜i j ) = (a ji ). Given matrices A ∈ F p×m , B ∈ Fm×n , we define the product AB ∈ F p×n of the matrices by (AB)i j = m ∑ aik bk j . , we have A(BC) = (AB)C, A(B1 + B2 ) = AB1 + AB2 , (A1 + A2)B = A1 B + A2B. 2) In Fn×n we define the identity matrix In by In = (δi j ).

An (z) ∈ F[z] such that n ∑ ai (z)pi (z) = 1. 15), one of the most important equations in mathematics, will be refered to as the Bezout equation. The importance of polynomials in linear algebra stems from the strong connection between factorization of polynomials and the structure of linear transformations. The primary decomposition theorem is of particular applicability. 39. A polynomial p(z) ∈ F[z] is factorizable, or reducible, if there exist polynomials f (z), g(z) ∈ F[z] of degree ≥ 1 such that p(z) = f (z)g(z).

2 Vector Spaces 35 (ai j ) + (bi j ) = (ai j + bi j ), α (ai j ) = (α ai j ). These definitions make Fm×n into a vector space. Given the matrix A = (ai j ), ˜ as the n × m matrix given by we define its transpose, which we denote by A, (a˜i j ) = (a ji ). Given matrices A ∈ F p×m , B ∈ Fm×n , we define the product AB ∈ F p×n of the matrices by (AB)i j = m ∑ aik bk j . , we have A(BC) = (AB)C, A(B1 + B2 ) = AB1 + AB2 , (A1 + A2)B = A1 B + A2B. 2) In Fn×n we define the identity matrix In by In = (δi j ).

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Spinc-manifolds with Pin.2/-action by Dessai A.


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