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Lecture Notes 8

# Lecture Notes 8 - Lecture08 Lecture 08 Vector Spaces and...

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Lecture08 1 September 27, 2007 Outline: 1) Quick Review of the Column Space and Null Space 2) Finding N(A) part 1: Gaussian Elimination to Upper Echelon Form "Special Solutions" the rank r of a matrix 3) Finding N(A) part 2: Gauss-Jordan Elimination to Reduced Row Echelon Form (RREF) Examples 4) Existence and Uniqueness: The general solution to Ax =b Lecture 08: Vector Spaces and Subspaces #2: The Null Space N(A) Review: The Column Space C(A) Definition: The Column Space of a Matrix A is the vector subspace formed by all linear combinations of the columns of A (general mxn matrix) 1) C(A) is Ax when x assumes all values in R n 2) C(A) is a subspace of R 3) C(A) controls the ______________________ of solutions to Ax =b 4) Ax =b will only have solutions iff b _________________________ Review: The Null Space N(A) Definition: The Null Space of a Matrix A is the vector subspace formed by all solutions x to Ax =0 (i.e. is all linear combination of columns of A that cancel to zero) 1) N(A) is a subspace of R 2) N(A) controls the ______________________ of solutions to Ax =b 3) If A is invertible, then N(A)=______________

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