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**Unformatted text preview: **Math 205, Spring 2011 Homework 5: due Feb 21 Chapter 3, Section 4; Chapter 4, Sections 1, 2. Week 5: 4.1, 4.2: R n and Vector Spaces ———– 1. Vector addition; scalar multiplication in R 2 . Vectors are vectorx = ( 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: vector 0 = (0 , 0) . additive inverse:- vectorx =- ( x, y ) = (- x,- y ) . distributive rules (2) ———- standard unit vectors: vector i = (1 , 0) , vector j = (0 , 1) . Linear combination property: vectorx = ( x, y ) = ( x, 0) + (0 , y ) = x vector i + y vector j. R 3 : vectorx + vector y, k · vectorx, vector 0 = (0 , , 0);- vectorx =- ( x, y, z ) = (- x,- y,- z ) . standard unit vectors vector i, vector j, vector k. vectorx = ( x, y, z ) = ( x, , 0) + (0 , y, 0) + (0 , , z ) = x vector i + y vector j + z vector k. 2 R n : vectorx + vector y = ( x 1 , x 2 , . . . , x n ) + ( y 1 , y 2 , . . . , y n ) = ( x 1 + y 1 , . . . , x n + y n ) . k · vectorx = k ( x 1 , x 2 , . . . , x n ) = ( kx 1 , . . . , kx n ) vector 0 = (0 , , . . . , 0) ,- vectorx =- ( x 1 , x 2 , . . . , x n ) = (- x 1 ,- x 2 , . . . ,- x n ) ....

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