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**Unformatted text preview: **MA 35100 LECTURE NOTES: MONDAY, APRIL 26 Dot Product For vectors ~v, ~w R n , we defined the dot product as the scalar ~v ~w = v 1 w 1 + v 2 w 2 + + v n w n in terms of ~v = v 1 v 2 . . . v n , ~w = w 1 w 2 . . . w n . With the dot product, we are able to define and prove many things: Length: Since ~v ~v 0, define || ~v || = ~v ~v . Cauchy-Bunyakowsky-Schwarz Inequality: ( ~v ~w ) 2 ( ~v ~v )( ~w ~w ) . Angles between Vectors: = arccos ~v ~w || ~v |||| ~w || . Orthogonal Complement: Let V be a subspace of R n , and define V = { ~x R n : ~v ~x = 0 for all ~v V } . Orthonormal Basis: If V has basis A = ( ~v 1 ,~v 2 ,...,~v n ), we can construct a basis B = ( ~u 1 ,~u 2 ,...,~u n ) of V which satisfies: ~u i ~u j = ( 1 if i = j , 0 otherwise. Orthogonal Projection: The transformation proj V : R n R n defined by proj V ~x = ( ~x ~u 1 ) ~u 1 + ( ~x ~u 2 ) ~u 2 + + ( ~x ~u n ) ~u n satisfies im(proj V ) = V and ker(proj V ) = V . We generalize the notion of a dot product to see which of these results can be generalized. Inner Product Spaces Recall the following definition: Definition. A linear space is a set V that is closed under linear combinations. More precisely, given f, g, h V and scalars c, k R , the following rules are satisfied: Closure: f + g V and k f V ....

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