s08hw4 - Week 4: 1.6 Properties of Determinants 2.1 R n and...

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Unformatted text preview: Week 4: 1.6 Properties of Determinants 2.1 R n and Vector Spaces 2.2 Subspaces/Spanning 2 1. Vector addition; scalar multiplication in R 2 . Vectors are x = ( x, y ) , with x, y real numbers. ( x 1 , y 1 ) + ( x 2 , y 2 ) = ( x 1 + x 2 , y 1 + y 2 ). parallelogram law: (2 , 1) + (1 , 4) = (3 , 5) (picture!) k ( x, y ) = ( kx, ky ) , scaling factor 3(2 , 1) = (6 , 3);- 1 2 (2 , 1) = (- 1 ,- 1 2 ) . (sketch) zero vector: 0 = (0 , 0) . additive inverse:- x =- ( x, y ) = (- x,- y ) . distributive rules (2) standard unit vectors: i = (1 , 0) , j = (0 , 1) . Linear combination property: x = ( x, y ) = ( x, 0) + (0 , y ) = xi + yj. R 3 : x + y, k x, 0 = (0 , , 0);- x =- ( x, y, z ) = (- x,- y,- z ) . standard unit vectors i, j, k. x = ( x, y, z ) = ( x, , 0) + (0 , y, 0) + (0 , , z ) = xi + yj + zk. 4 R n : x + y = = ( x 1 , x 2 , . . . , x n ) + ( y 1 , y 2 , . . . , y n ) ( x 1 + y 1 , . . . , x n + y n ) . k x = k ( x 1 , x 2 , . . . , x n ) = ( kx 1 , . . . , kx n ) 0 = (0 , , . . . , 0) ,- x =- ( x 1 , x 2 , . . . , x n ) = (- x 1 ,- x 2 , . . . ,- x n ) . V any collection, elements v V called vectors. Formula for vector plus and scalar mult. Usually scalars are real k R ; may also have k C , complex. ( V , + , ) is a Vector Space if the vector + and scalar mult satisfy 10 rules: 2 closure rules 4 rules for + (including 0; and- x ) 2 rules for (including 1 v = v ) 2 distributive laws. (2+4+2+2 = 10 or 2+8.) 6 Examples: R n ; V = M 2 2 ( R ); F ( a, b ) = functions: f+g, k f, 0, -f....
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s08hw4 - Week 4: 1.6 Properties of Determinants 2.1 R n and...

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