section41 - Chapter 4: The Vector Space Rn MAT188H1F Lec03...

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Chapter 4: The Vector Space R n MAT188H1F Lec03 Burbulla Chapter 4 Lecture Notes Fall 2007 Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 4: The Vector Space R n Chapter 4: The Vector Space R n 4.1: Subspaces of R n and Spanning Sets Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla
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Chapter 4: The Vector Space R n 4.1: Subspaces of R n and Spanning Sets Introduction In Chapter 4 we move from R 3 to R n , which immediately makes things more abstract, since you can’t visualize higher dimensions. In addition, Chapter 4 introduces many new concepts: subspace, spanning sets, span of a set, linear independence, linear dependence, basis, dimension, null space, row space, column space, orthogonal complements, orthonormal basis, orthogonal matrix, . .. to mention but a few! The good news is that there are no new computational techniques involved. Everything we shall do in Chapter 4 comes down to either solving a system of equations, especially homogeneous systems of equations; reducing a matrix; finding a determinant; adding or subtracting vectors; multiplying vectors by a scalar; multiplying matrices; or calculating dot products. There are well-defined algorithms for everything we shall do. It’s just that sometimes you may lose sight of why you are doing certain calculations. The payoff will be some very important results, cf. Sections 4.6 and 4.7. Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 4: The Vector Space R n 4.1: Subspaces of R n and Spanning Sets Set Notation Since we will be talking a lot about sets of vectors, here’s a quick review (?) of set theory notation. I x A is read “ x is an element of the set A , ” or “ x is in A . I A = { x , y , z , . . . } is a list of the elements in A . I A = { x B | P ( x ) is true } is read “ A is the set of x in B such that the property P ( x ) is true.” I φ is the empty set, the set with no elements. I For example, in calculus, [ a , b ] = { x R | a x b } is set notation for the closed interval [ a , b ]; it means “the set of x in R such that a x b . I For example, in linear algebra, { X R 3 | AX = B } is the set of solutions in R 3 to the system of linear equations AX = B . Chapter 4 Lecture Notes MAT188H1F Lec03 Burbulla
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Chapter 4: The Vector Space R n 4.1: Subspaces of R n and Spanning Sets The Vector Space R n For given n , the set of n × 1 column matrices R n = x 1 x 2 . . . x n ± ± ± ± ± ± ± ± each x i R , together with the usual matrix addition and scalar multiplication x 1 x 2 . . . x n + y 1 y 2 . . . y
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This note was uploaded on 06/22/2008 for the course ENGINEERIN MAT188 taught by Professor Burbula during the Fall '07 term at University of Toronto- Toronto.

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section41 - Chapter 4: The Vector Space Rn MAT188H1F Lec03...

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