s08week04

s08week04 - Math 205, Spring 2008 B. Dodson Week 4: 1.6...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Math 205, Spring 2008 B. Dodson Week 4: 1.6 Properties of Determinants 2.1 R n and Vector Spaces 2.2 Subspaces/Spanning ———– 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 ) = x ± i + y ± j. R 3 : ±x + ± y, k · ±x, ± 0 = (0 , 0 , 0); - ±x = - ( x, y, z ) = ( - x, - y, - z ) . standard unit vectors ± i, ± j, ± k. ±x = ( x, y, z ) = ( x, 0 , 0) + (0 , y, 0) + (0 , 0 , z ) = x ± i + y ± j + z ± k.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 , . . . , 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.) ——- Examples: R n ; V = M 2 × 2 ( R ); F ( a, b ) = functions: f+g, k · f, 0, -f. P
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 7

s08week04 - Math 205, Spring 2008 B. Dodson Week 4: 1.6...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online