2.1 Linear Transformations 3

# 2.1 Linear Transformations 3 - 2.1 Say dim V = n 2 cfw_v1...

This preview shows pages 1–4. Sign up to view the full content.

2.1. * Say dim V = n 2 { v 1 , v 2 , · · · , v m } : a basis for W { v 1 , v 2 , · · · , v m , v m +1 , · · · , v n } : a basis for V Let W 0 = span { v m +1 , · · · , v n } and W 00 = span { v m +1 - v 1 , v m +2 - v 3 , · · · , v n - v 5 } then V = W W 0 = W W 00 Clearly W 0 6 = W 00 ( ) If W 0 = W 00 , then v m +1 , v m +1 - v 1 W 00 v 1 W W 0 = (0) It’s a contradiction (b) Example (example 1) the projection on W along W 0 1 { ( a, b )) = (0 , b - 1 3 a ) + ( a, 1 3 a ) } (example 2) the projection on W along W 0 2 ( a, b ) = (0 , b ) + ( a, 0) 28. (1) { 0 } is T -invariant x ∈ { 0 } , T ( x ) = 0 ∈ { 0 } ( T is linear) (2) V is T -invariant (T(V) V) x V , T ( x ) V ( T : V V ) (3) R ( T ) is T -invariant (T(T(V)) T(V)) 68 PNU-MATH

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

View Full Document
2.1. , T ( x ) R ( T ), T ( T ( x )) T ( V ) R ( T ) (4) N ( T ) is T-invariant x N ( T ), T( x ) = 0 N ( T ) ( N ( T ) V as subspace, so N ( T ) has zero) 29. If T ( W ) W , x, y W, and c F T W ( x + y ) = T ( x + y ) = T ( x ) + T ( y ) = T W ( x ) + T W ( y ) (W is T-invariant and T is linear) T W ( cx ) = T ( cx ) = cT ( x ) = cT W ( x ) T W is linear 30. x V, T ( x ) = x 1 s.t x = x 1 + x 2 , x 1 W, x 2 W 0 (1) W is T -invariant x 1 W , T( x 1 ) = x 1 W T( W ) W (2) T W = I W T W : W W, x 1 W, T W ( x 1 ) = x 1 I W : W W, x 1 W, I W ( x 1 ) = x 1 x 1 W, T W ( x 1 ) = I W ( x 1 ) T W = I W 31. V = R ( T ) W , W is T-invariant 69 PNU-MATH
2.1. (a) T ( W ) T ( V ) W = (0) T ( W ) = 0 T ( W ) N ( T ) (b) Since V = R ( T ) W , so dim V = dim R ( T ) + dim W rank ( T ) + nullity ( T ) = dim V Since dim V < , dim W = nullity ( T ) (c) (Example 1) Exercise 21, left shift (Example 2) β = { v 1 , v 2 , · · · } for V T : V V, T ( v i ) = 0 if i is odd i 2 if i is even Then R ( T ) = V, N ( T ) = span ( { v 1 , v 3 , v 5 , · · · } ) , W = (0) V = R ( T ) W (Example 3) dim V = 0 , { v 1 , v 2 , v 3 , v 4 , v 5 , v 6 , · · · } : a basis for V { v 1 , v 2 , v 3 , v 5 , v 6 , v 7 , v 9 , v 10 , · · · } : a basis for R ( T ) { v 3 , v 4 , v 7

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.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern