College Algebra Exam Review 351

College Algebra Exam Review 351 - .1/;1 a .n/;n : Moreover,...

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8.3. MULTILINEAR MAPS AND DETERMINANTS 361 of elements of R n , and denote the i –th entry of a j by a i;j , as above, Then a j D P i a i;j O e i . By the multilinearity of ± , ±.a 1 ;a 2 ;:::;a n / D ±. X i 1 a i 1 ;1 O e i 1 ;:::; X i n a i n ;n O e i n / D X i 1 ;i 2 ;:::;i n a i 1 ;1 ±±± a i n ;n ±. O e i 1 ;:::; O e i n /: Because ± is alternating, ±. O e i 1 ;:::; O e i n / is zero unless the sequence of indices .i 1 ;:::;i n / is a permutation of .1;2;:::;n/ . Thus ±.a 1 ;a 2 ;:::;a n / D X ± 2 S n a ±.1/;1 ±±± a ±.n/;n ±. O e ±.1/ ;:::; O e ±.n/ D X ± 2 S n a ±.1/;1 ±±± a ±.n/;n ².³/±. O e 1 ;:::; O e n / D ´.a 1 ;:::;a n /±. O e 1 ;:::; O e n /: We have proved the following result: Proposition 8.3.6. There is a unique alternating multilinear function ´ W .R n / n ²! R satisfying ´. O e 1 ;:::; O e n / D 1 . The function ´ satisfies ´.a 1 ;:::;a n / D X ± 2 S n ².³/a 1;±.1/ ±±± a n;±.n/ D X ± 2 S n ².³/a
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Unformatted text preview: .1/;1 a .n/;n : Moreover, if W .R n / n ! N is any alternating and multilinear function, then for all a 1 ;:::;a n 2 R n , .a 1 ;:::;a n / D .a 1 ;:::;a n /. O e 1 ;:::; O e n /: Denition 8.3.7. The determinant of an n by n matrix with entries in R is dened by det .A/ D .a 1 ;:::;a n /; where a 1 ;:::;a n 2 R n are the columns of A . Corollary 8.3.8. (a) The determinant is characterized by the following properties: (i) det .A/ is an alternating multilinear function of the columns of A ....
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