Review3

# Review3 - W and z ⊥ is orthogonal to W . • Be able to...

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Review Sheet 3 - Math 291 -last third of course Definitions: Inner product, Inner products expressed as < x , y > G = x t G y , positive - deﬁnite inner products, the dot product, orthogonal vectors, orthonormal basis, orthogonal complement of a subspace, orthogonal projection onto a subspace, norm, a least-squares solution to A x = y , the normal equation. Computations (for simple, low-dimensional cases) Normalize vectors. Find the coordinates of v in an orthonormal basis. Given a basis, use the Gram-Schmidt algorithm to construct an orthonormal basis. Know how to compute Π W ( v ) , where Π W is the orthogonal projection onto W , and v is an arbitrary vector. In particular, if z is some vector and W is the subspace span( z ), be able to write z = z || + z , where z || lies in
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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 Cauchy-Schwarz inequality; Pythagoras’ theorem. • Examples of positive-deﬁnite and non-positive 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.

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