# lecture_38 - MA 35100 LECTURE NOTES: MONDAY, APRIL 26 Dot...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

## This note was uploaded on 03/31/2012 for the course MA 351 taught by Professor ?? during the Spring '08 term at Purdue University-West Lafayette.

### Page1 / 4

lecture_38 - MA 35100 LECTURE NOTES: MONDAY, APRIL 26 Dot...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online