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Unformatted text preview: Chapter 4: The Vector Space R n MAT188H1F Lec0106 Burbulla Chapter 4 Lecture Notes Fall 2010 Chapter 4 Lecture Notes MAT188H1F Lec0106 Burbulla Chapter 4: The Vector Space R n Chapter 4: The Vector Space R n 4.1: Subspaces of R n and Spanning Sets 4.2: Linear Independence 4.3: Basis and Dimension 4.4: Rank 4.5: Orthogonality 4.6: Projections and Approximation 4.7: Orthogonal Diagonalization Chapter 4 Lecture Notes MAT188H1F Lec0106 Burbulla Chapter 4: The Vector Space R n 4.1: Subspaces of R n and Spanning Sets 4.2: Linear Independence 4.3: Basis and Dimension 4.4: Rank 4.5: Orthogonality 4.6: Projections and Approximation 4.7: Orthogonal Diagonalization Introduction In Chapter 4 we move from R 3 to R n , which immediately makes things more abstract, since you cant 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. Chapter 4 Lecture Notes MAT188H1F Lec0106 Burbulla Chapter 4: The Vector Space R n 4.1: Subspaces of R n and Spanning Sets 4.2: Linear Independence 4.3: Basis and Dimension 4.4: Rank 4.5: Orthogonality 4.6: Projections and Approximation 4.7: Orthogonal Diagonalization Set Notation Since we will be talking a lot about sets of vectors, heres 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 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 Lec0106 Burbulla Chapter 4: The Vector Space R n 4.1: Subspaces of R n and Spanning Sets 4.2: Linear Independence 4.3: Basis and Dimension 4.4: Rank 4.5: Orthogonality 4.6: Projections and Approximation 4.7: Orthogonal Diagonalization The Vector Space R n For given n , the set of n 1 column matrices R n = x 1 x 2 .. ....
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This note was uploaded on 12/23/2010 for the course MAT mat 188 taught by Professor Dietrichburbulla during the Spring '10 term at University of Toronto Toronto.
 Spring '10
 DietrichBurbulla
 Linear Algebra, Algebra, Vector Space, Sets

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