# ass7 - MATH 223, Linear Algebra Fall, 2007 Solutions to...

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

MATH 223, Linear Algebra Fall, 2007 Solutions to Assignment 7 1. A linear operator T on a vector space V is called a projection if T 2 = T . (We will be looking at orthogonal projections later.) (a) If T = T A is represented by the matrix A = 2 0 2 0 1 0 - 1 0 - 1 , show that T is a projection. Solution: Since T 2 will be represented by A 2 , it’s enough to show that A 2 = A . I skip this very easy calculation. (b) Show that, if T is a projection on V , then so is I - T ; here I is the identity operator on V . Solution: For any ~v V , ( I - T ) 2 ~v = ( I - T )( I - T ) ~v = ( I - T )( ~v - T~v ) = ~v - T~v - T ( ~v - T~v ) = ~v - T~v - T~v + T 2 ~v. Using T 2 = T , this reduces to ~v - T~v = ( I - T ) ~v . So ( I - T ) 2 = I - T and I - T is a projection, too (if T is). (c) Show that, if T is a projection on V , then V = ker ( T ) im ( T ). (Recall that is the direct sum, so that not only do we have V = ker ( T ) + im ( T ) but also, ker ( T ) im ( T ) = ~ 0 } .) Solution: For any ~v V , of course T~v im ( T ). Now T ( ~v - T~v ) = T~v - T 2 ~v = ~ 0, so ~v - T~v ker ( T ). As ~v = ( ~v - T~v ) + T~v , we have ~v ker ( T ) + im ( T ) and so V = ker ( T ) + im ( T ). To show the sum is direct, suppose that ~v ker ( T ) im ( T ). Then T~v = ~ 0 and also ~v = T ~w for some ~w V . Then ~ 0 = T~v = T ( T ~w ) = T 2 ~w = T ~w = ~v . So ker ( T ) im ( T ) = { ~ 0 } , and thus V = ker ( T ) im ( T ). (d) Show that, if T is a projection on V , its only possible eigenvalues are 0 and 1. Solution: One can do this directly, using the deﬁnition of eigenvalues. Supposing that

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.

## This note was uploaded on 04/29/2008 for the course MATH 223 taught by Professor Loveys during the Fall '07 term at McGill.

### Page1 / 4

ass7 - MATH 223, Linear Algebra Fall, 2007 Solutions to...

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

View Full Document
Ask a homework question - tutors are online