lecture11

# lecture11 - Lecture 11 Fundamental Theorems of Linear...

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Lecture 11 Fundamental Theorems of Linear Algebra Orthogonalily and Projection Shang-Hua Teng

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The Whole Picture Rank(A) = m = n A x = b has unique solution [ ] I R = [ ] F I R = = 0 I R = 0 0 F I R Rank(A) = m < n A x = b has n-m dimensional solution Rank(A) = n < m A x = b has 0 or 1 solution Rank(A) < n, Rank(A) < m A x = b has 0 or n-rank(A) dimensions
Basis and Dimension of a Vector Space A basis for a vector space is a sequence of vectors that The vectors are linearly independent The vectors span the space: every vector in the vector can be expressed as a linear combination of these vectors

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Basis for 2D and n-D (1,0), (0,1) (1 1), (-1 –2) The vectors v 1 ,v 2 ,…v n are basis for R n if and only if they are columns of an n by n invertible matrix
Column and Row Subspace C(A): the space spanned by columns of A Subspace in m dimensions The pivot columns of A are a basis for its column space Row space: the space spanned by rows of A Subspace in n dimensions The row space of A is the same as the column space of A T , C(A T ) The pivot rows of A are a basis for its row space The pivot rows of its Echolon matrix R are a basis for its row space

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Important Property I: Uniqueness of Combination The vectors v 1 ,v 2 ,…v n are basis for a vector space V, then for every vector v in V, there is a unique way to write v as a combination of v 1 ,v 2 ,…v n .
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