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

This preview shows pages 1–7. Sign up to view the full content.

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

View Full Document

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

View Full Document

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

{[ snackBarMessage ]}

Page1 / 17

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

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online