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

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

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

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