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## S. E. Payne's A Second Semester of Linear Algebra PDF

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By S. E. Payne

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Then we have [(T ◦ S)(u)]B3 = [T (S(u))]B3 = [T ]B3 ,B2 [S(u)]B2 = [T ]B3 ,B2 [S]B2 ,B1 [u]B1 = = [T ◦ S]B3 ,B1 [u]B1 for all u ∈ U. This implies that [T ◦ S]B3 ,B1 = [T ]B3 ,B2 · [S]B2 ,B1 . This is the equation that suggests that the subscript on the matrix representing a linear map should have the basis for the range space listed first. Recall that L(U, V ) is naturally a vector space over F with the usual addition of linear maps and scalar multiplication of linear maps. Moreover, for a, b ∈ F and S, T ∈ L(U, V ), it follows easily that [aS + bT ]B2 ,B1 = a[S]B2 ,B1 + b[T ]B2 ,B1 .

But also we have the distributive properties (S1 + S2 )T = S1 T + S2 T and S(T1 + T2 ) = ST1 + ST2 whenever the products are defined. In general the multiplication of linear maps is not commutative even when both products are defined. 2 Kernels and Images We use the following language. If f : A → B is a function, we say that A is the domain of f and B is the codomain of f . 2. KERNELS AND IMAGES 33 B. We define the image (or range of f by Im(f ) = {b ∈ B : f (a) = b for at least one a ∈ A}. And we use this language for linear maps also.

Pk are the distinct monic primes occurring in this factorization of f , then f = pn1 1 pn2 2 · · · pnk k , where ni is the number of times the prime pi occurs in this factorization. This decomposition is also clearly unique and is called the primary decomposition of f (or simply the prime factorization of f ). 7. Let f be a non-scalar monic polynomial over the field F , and let f = pn1 1 · · · pnk k be the prime factorization of f . For each j, 1 ≤ j ≤ k, let fj = f pj nj pni i . = Then f1 , .