By C. H. Edwards Jr.

ISBN-10: 0122325508

ISBN-13: 9780122325502

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**Extra info for Advanced Calculus of Several Variables**

**Sample text**

Let the matrices A, B, C be of dimensions k x /, / x m, and m x n respectively. Then let f:&l^&k,g: Mm -► M\ /? : &n -► Mm be the linear maps such that Mf = A,Mg = B, Mh = C. Then (/o(0oA))(x) =f(goh(x)) = f(9(h)x)))=(fog)(h(x)) = ((fog)oh)(x) n for all x e M , so f° (g ° h) = (f ° g) ° h. 2 therefore implies that A(BC) = MfMgûh = Mfo(goh) = (MfMg)Mh = (AB)C, thereby verifying associativity. To prove (b), let A be an / x m matrix, and B, C m x n matrices. Then let / : &m -> St1 and g, h : 0tn -+ $m be the linear maps such that Mf = A, Mg = B, and Mh = C.

A„), (4) 36 I Euclidean Space and Linear Mappings Z)(a1? . , a4 + ra,·, . . , an) = Z)(al5 . . , a,, . . , a„), (5) and D ( a 1 ? . , a,·, . . , a i5 . . , a„) = - / ) ( a l 5 . . , a t , . . , a y , . . y. , An are as usual the column vectors of A. Then (4) above says that the determinant of A is multiplied by r if some column of A is multiplied by r, (5) that the determinant of A is unchanged if a multiple of one column is added to another column, while (6) says that the sign of det A is changed by an interchange of any two columns of A.

5 If A is a compact subset of Mm and i? is a compact subset of 9tn9 then ,4 x £ is a compact subset of @m+n. PROOF Given a sequence {c,}f ={(a f , b,)}f of points of A x B, we want to show that it has a subsequence converging to a point of A x /?. ^ has a subsequence {a^JLi which converges to a point Sie A. }*= i has a subsequence {b(/;k)}^= { which converges to a point be B. = { is a subsequence of the original sequence {(af, b , ) ^ } which converges to the point (a, b) e A x B. | We are now ready for the criterion that will serve as our recognition test for compact sets.

### Advanced Calculus of Several Variables by C. H. Edwards Jr.

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