Study Guides (238,597)
Mathematics (538)
MAT224H1 (48)
Sean Uppal (42)

# Summary notes 3End

9 Pages
187 Views

School
University of Toronto St. George
Department
Mathematics
Course
MAT224H1
Professor
Sean Uppal
Semester
Summer

Description
MAT224H1c.doc Inner Product Spaces INNER P RODUCTS Definition An inner producton a vector space V is a function that assigns a numberv,w to every pair v, w of vector space V in such a way that the following axioms holds: P 1 v,w is a real number. P 2: v,w = w,v . P 3 v+ w,u = v,u + w,u . P 4 rv,w = r v,w . P 5 v,v 0,vV . Definition A vector space V with an inner product is called an inner product space. Note , :V V R . V,R,+, ) is a vector space. V,R,+,, , ) is an inner product space. Examples The following are inner product spaces. 1) (R ,R,+,, , ), define X,Y = X Y the dot product. 2) C [a,b],R,+,, , , define f , g = bf(x ) x(dx). a T 3) (M mn,R,+,, , , define A, B = tr AB . Theorem Let , be an inner product on a space V. Let u, v, w denote vectors in V, r a real number. Then: 1) u,v + w = u,v + u, w . 2) v,rw = r v,w . 3) v,0 = 0 = 0,v . 4) v,v = 0 if and only if v = 0 . Theorem T n n If A is any nn positive definite matrix, thX,Y = X AY,X,Y R defines an inner product on R, and every inner product on R , and every inner product on R arises in this way. Proof: Page 1 of 9 www.notesolution.com
More Less

Related notes for MAT224H1

OR

Don't have an account?

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Join to view

OR

By registering, I agree to the Terms and Privacy Policies
Already have an account?
Just a few more details

So we can recommend you notes for your school.