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# Review_L1234 - MA2213 Review Lectures 1-4 Inner Products...

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Unformatted text preview: MA2213 Review Lectures 1-4 Inner Products Gramm Matrices Gram-Schmidt Orthogonalization Transpose and its Properties Transpose ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − π π 2 3 7 4 7 2 3 4 8 3 8 3 T T T T M N MN = ) ( T T M M M ) ( ) ( det 1 1 − − = ⇒ ≠ nd M M T det det = heorem 1 M M T is positive definite I I MM M M T T T T = = = − − ) ( ) ( 1 1 roofs ) ( ) ( ) ( > = ⇒ ≠ Mv Mv v M M v v T T T Inner ( = Scalar) Product Spaces is a vector space R V v u R v u ∈ ∈ , , ) , ( ymmetry V over reals ith an inner product that satisfies the following 3 properties: ) , ( ) , ( u v v u = linearity V w v u R b a w u b v u a bw av u ∈ ∈ + = + , , , , ), , ( ) , ( ) , ositivity ) , ( > ⇒ ≠ u u u emark Symmetry and Linearity imply ) , ( ) , ( ) , ( ) , ( ) , ( ) , u w b u v a w u b v u a bw av u u bw av + = + = + = + ence (- , -) : V x V Æ R is Bilinear Examples of Inner Product Spaces xample 2....
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Review_L1234 - MA2213 Review Lectures 1-4 Inner Products...

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