Math121sol3

# Math121sol3 - Math 121 Linear Algebra and Applications Prof...

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: Math 121: Linear Algebra and Applications Prof. Lydia Bieri Solution Set 3 Posted: Tue. Oct. 30, 2007 Written by: Luca Candelori Exercise 1 (1.6/35) . (a) Let v + W be a typical element of V/W . Since v ∈ V , we can write it as v = a 1 u 1 + ... + a k u k + a k +1 u k +1 + ... + a n u n Since u 1 ,...u k are all in W , in V/W this becomes v + W = a k +1 u k +1 + ... + a n u n + W which shows that the set { u k +1 + W,...,u n + W } spans V/W . To prove linear idependence, suppose there are scalars a k +1 ,...,a n such that w = a k +1 u k +1 + ... + a n u n ∈ W then we can write w as a linear combination of the u 1 ,...,u k , since they form a basis for W . But this would violate linear independence of the u 1 ,...,u n , since they form a basis for V . (b) We know that dim W = k and dim V = n so that there are n- k elements in the basis for V/W provided. Therefore dim( V/W ) = dim V- dim W Exercise 2 (2.1/5) . We prove first T is linear. Suppose we have polynomials f,g ∈ P 2 ( R ). Then T ( f ( x ) + g ( x )) = x ( f ( x ) + g ( x )) + ( f ( x ) + g ( x )) = xf ( x ) + xg ( x ) + f ( x ) + g ( x ) = ( xf ( x ) + f ( x )) + ( xg ( x ) + g ( x )) = T ( f ( x )) + T ( g ( x )) Also, for any real number c , we have T ( cf ( x )) = xcf ( x ) + ( cf ( x )) = c ( xf ( x ) + f ( x ) ) = cT ( f ( x )) Write now f ( x ) = ax 2 + bx + c , with a,b,c ∈ R , a typical element of P 2 ( R ). For it to be in N ( T ) we need T ( f ( x )) = 0, or, in other words x (...
View Full Document

## This note was uploaded on 10/23/2010 for the course MATH Math 121 taught by Professor Bieri during the Fall '07 term at Harvard.

### Page1 / 5

Math121sol3 - Math 121 Linear Algebra and Applications Prof...

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

View Full Document
Ask a homework question - tutors are online