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Lecture Notes 10

# Lecture Notes 10 - Lecture10 Lecture 10 The Big picture...

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Lecture10 October 03, 2007 Outline: 1) Bases and Dimensions for the 4 Fundamental Subspaces of a m x n Matrix A The Column Space C(A) The Null Space N(A) The Row Space C(A T ) The Left Null Space N(A T ) 2) Orthogonality of the 4 subspaces 3) The full picture of Ax =b Lecture 10: The Big picture cont'd Basis and Dimension of the 4 subspaces of a matrix A Given A in R m x n there are four fundamental subspaces associated with the matrix A Name Symbol Subspace Dimension 1) Column Space 2) Null Space 3) Row Space 4) Left Null Space Basis and Dimension of the 4 subspaces of a matrix A Example: A= = LR= 1 2 3 0 1 1 2 3 1 3 2 4 6 -2 -2 1 2 3 0 1 0 0 0 1 2 0 0 0 0 0 1 0 0 1 1 0 2 -2 1 The Column Space C(A): Controls _____________ of solutions to Ax =b Subspace of Dimension: Basis: Comments: C(A)___C(R) Basis and Dimension of the 4 subspaces of a matrix A Example: A= = LR= 1 2 3 0 1 1 2 3 1 3 2 4 6 -2 -2 1 2 3 0 1 0 0 0 1 2 0 0 0 0 0 1 0 0 1 1 0 2 -2 1 The Null Space C(A): Controls _____________ of solutions to Ax =b Subspace of Dimension: Basis: Comments: N(A)___N(R)

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Lecture10 October 03, 2007
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Lecture Notes 10 - Lecture10 Lecture 10 The Big picture...

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