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Unformatted text preview: W and z ⊥ is orthogonal to W . • Be able to solve the normal equation A t A x = A t y to ﬁnd least squares solutions. Facts, theorems, etc. – know how to prove, and how to use: • The CauchySchwarz inequality; Pythagoras’ theorem. • Examples of positivedeﬁnite and nonpositive deﬁnite inner products • Let f A : R n → R m be the linear transformation given by f A ( x ) = A x . Then – Range( f A ) = Column space of A . – Range( f A t ) = Row space of A . – Ker( A t ) = [Range( f A )] ⊥ = [col space of A] ⊥ . – Ker( A ) = [Range( f A t )] ⊥ = [row space of A] ⊥ . • ˜ x is a least squares solution to A x = y ⇐⇒ A t A ˜ x = A t y . 1...
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This note was uploaded on 05/23/2008 for the course MATH 290/291 taught by Professor Lerner during the Spring '08 term at Kansas.
 Spring '08
 LERNER
 Linear Algebra, Algebra, Vectors, Dot Product

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