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Lecture #2 Notes

# Lecture #2 Notes - Matrix Theory Math6304 Lecture Notes...

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Matrix Theory, Math6304 Lecture Notes from August 30, 2012 taken by Andy Chang Last Time (8/28/12) Course info: website - math.uh.edu/ bgb Matrix multiplication: left (premultiplication) and right (postmultiplication) Nullity and rank: ”dimension counting” Dot product and orthogonality: inner product, orthogonal projection, least squares property Gram-Schmidt: - set of linearly independent vectors can yield an orthonormal set that spans the space 1 Further Review 1.1 Gram-Schmidt (cont’d) 1.1.9 Proposition (Gram-Schmidt) . Given a linearly independent set { x 1 , x 2 , . . . , x n } in C m , then there exists an orthonormal system { z 1 , z 2 , . . . , z n } such that for each j n , span { x l : l j } = span { z l : l j } . Proof. Since orthonormality implies linear independence, the dimension of both sides is equal. It is enough to show that we can find inductively for each j n a vector z j which forms an orthonormal system { z 1 , z 2 , . . . , z j } with the preceding ones and z j = u j, 1 x 1 + u j, 2 x 2 + · · · + u j,j x j , with appropriate coe cients u j,k , so z j span { x l : l j } . As a consequence, z k span { x l : l j } for k j and thus span { z l : l j } span { x l : l j } but since the dimension on both sides is equal, the two spans must be the same.

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Lecture #2 Notes - Matrix Theory Math6304 Lecture Notes...

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