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# Chapter4 - Chapter 4 The Vector Space R n MAT188H1F Lec0106...

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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 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. 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, 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 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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Chapter4 - Chapter 4 The Vector Space R n MAT188H1F Lec0106...

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