asn1-soln - MAT237Y Multivariable Calculus Summer 2009...

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

View Full Document Right Arrow Icon
MAT237Y – Multivariable Calculus Summer 2009 Assignment # 1 Solutions. Questions are worth 10 marks each. Due Tuesday, May 19, at 6:10pm sharp. 1. (a) Using the dot product, show that for x,y R n , the formula 2 k x k 2 + 2 k y k 2 = k x + y k 2 + k x - y k 2 holds. (b) The norm on R n can be defined in terms of the dot product by the formula k x k = x x . Show that the reverse is true. That is, find a formula for x y involving the norms of vectors ( k x k , k y k , k x + y k , and k x - y k for example), and without using coordinates. Solution. (a) Consider ( x + y ) ( x + y ). By definition of the norm, this is equal to k x + y k 2 . But, from the properties of the dot product, we know the following: ( x + y ) ( x + y ) = x x + 2( x y ) + y y = k x k 2 + 2( x y ) + k y k 2 . So we have k x + y k 2 = k x k 2 + 2( x y ) + k y k 2 . (1) We can do a similar calculation to ( x - y ) ( x - y ), and we end up with the following equation: k x - y k
Background image of page 1

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

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 10/21/2010 for the course MATHEMATIC MAT237Y1 taught by Professor Romauldstanczak during the Fall '09 term at University of Toronto- Toronto.

Page1 / 4

asn1-soln - MAT237Y Multivariable Calculus Summer 2009...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online