lecture16

# lecture16 - Lecture 16 4 Multi-linear Algebra 4.1 Review of...

This preview shows pages 1–2. Sign up to view the full content.

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: Lecture 16 4 Multi-linear Algebra 4.1 Review of Linear Algebra and Topology In today’s lecture we review chapters 1 and 2 of Munkres. Our ultimate goal (not today) is to develop vector calculus in n dimensions (for example, the generalizations of grad, div, and curl). Let V be a vector space, and let v i ∈ V, i = 1 , . . . , k . 1. The v i s are linearly independent if the map from R k to V mapping ( c 1 , . . . , c k ) to c 1 v 1 + . . . + c k v k is injective. 2. The v i s span V if this map is surjective (onto). 3. If the v i s form a basis, then dim V = k . 4. A subset W of V is a subspace if it is also a vector space. 5. Let V and W be vector spaces. A map A : V → W is linear if A ( c 1 v 1 + c 2 v 2 ) = c 1 A ( v 1 ) + c 2 A ( v 2 ). 6. The kernel of a linear map A : V W is → ker A = { v ∈ V : Av = 0 } . (4.1) 7. The image of A is Im A = { Av : v ∈ V } . (4.2) 8. The following is a basic identity: dim ker A + dim Im A = dim V. (4.3) 9. We can associate linear mappings with matrices. Let...
View Full Document

## This note was uploaded on 10/02/2010 for the course MAT unknown taught by Professor Unknown during the Fall '04 term at MIT.

### Page1 / 3

lecture16 - Lecture 16 4 Multi-linear Algebra 4.1 Review of...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online