# 7.1 - But ±-5 2 3-1 ²± 1 2 3 5 ² M = IM = M so M = ± 0...

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Math 2940 Solutions, Fall 2011 Section 7 . 1 6 ) (a) Consider ~v = - ~w . Then T ( ~v + ~w ) = T ( ~ 0) is not even deﬁned, so the sum formula is not satisﬁed. For the scalar formula, T (2 ~v ) = 2 ~v/ || 2 ~v || = ~v/ || ~v || = T ( ~v ) 6 = 2 T ( ~v ) so this one fails too. (b) T ( ~v + ~w ) = v 1 + w 1 + v 2 + w 2 + v 3 + w 3 = v 1 + v 2 + v 3 + w 1 + w 2 + w 3 = T ( ~v ) + T ( ~w ). Similarly, T ( c~v ) = c ( v 1 + v 2 + v 3 ) = cT ( ~v ). (c) T ( ~v + ~w ) = v 1 + w 1 +2 v 2 +2 w 2 +3 v 3 +3 w 3 = v 1 +2 v 2 +3 v 3 + w 1 +2 w 2 +3 w 3 = T ( ~v )+ T ( ~w ). Similarly, T ( c~v ) = c ( v 1 + 2 v 2 + 3 v 3 ) = cT ( ~v ). (d) Note T ((1 , 2 , 3)) + T ((4 , 3 , 2)) = 3 + 4 = 7 while T ((1 , 2 , 3) + (4 , 3 , 2)) = T ((5 , 5 , 5)) = 5. Also, - 2 T ((1 , 2 , 3)) = - 6 while T (( - 2 , - 4 , - 6)) = - 2. 10 ) (a) Not invertible. T (1 , 0) = ~ 0. The kernel is nontrivial. (b) Not invertible. The range is the plane x + y - z = 0 which is not all of R 3 . (c) Not invertible. T (0 , 1) = ~ 0. 14 ) Notice that ± 1 2 3 5 ²± - 5 2 3 - 1 ² = ± 1 0 0 1 ² . Thus for any 2 × 2 matrix X , T ³± - 5 2 3 - 1 ² X ´ = ± 1 2 3 5 ²± - 5 2 3 - 1 ² X = IX = X so the range of T is all of V . Now suppose ± 1 2 3 5 ² M = ± 0 0 0 0 ² . Then ± - 5 2 3 - 1 ²± 1 2 3 5 ² M = ± - 5 2 3 - 1 ²± 0 0 0 0 ² = ± 0 0 0 0 ²
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Unformatted text preview: . But ±-5 2 3-1 ²± 1 2 3 5 ² M = IM = M so M = ± 0 0 0 0 ² . 18 ) T ³± a b c d ²´ = ± 1 0 0 0 ²± a b c d ²± 0 0 0 1 ² = ± a b 0 0 ²± 0 0 0 1 ² = ± b 0 0 ² so as long as the upper right entry is not 0, the image under T will not be zero. 31 ) Consider the vectors ~u and ~v deﬁning two sides of a rectangle. (The vertices are ~ 0, ~u , ~v and ~u + ~v .) Then the images of these four points under T , using the linearity are ~ 0, T ( ~u ), T ( ~v ) and T ( ~u )+ T ( ~v ), which form a parallelogram (squashed is { T ( ~u ) , T ( ~v ) } are dependent) by the rules of vector addition....
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## This note was uploaded on 01/02/2012 for the course MATH 2940 at Cornell.

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