2.1 Linear Transformations 3

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

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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
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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
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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
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