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Math416_Midterm3_sol - Math 416 Abstract Linear Algebra...

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Unformatted text preview: Math 416 - Abstract Linear Algebra Fall 2011, section E1 Midterm 3, November 16 Name: Solutions • No calculators, electronic devices, books, or notes may be used. • Show your work. No credit for answers without justification. • Good luck! 1. /10 2. /10 3. /10 4. /10 Total: /40 1 Problem 1. (10 pts) Let A : R 4 → R be the linear map defined by A x 1 x 2 x 3 x 4 = x 1 + x 2 + x 3 + x 4 . Find the projection of the vector v = 2- 1 3 onto the subspace ker A = { x ∈ R 4 | Ax = 0 } . Noting ker A = Span { w = 1 1 1 1 } ⊥ , the projection is Proj ker A ( v ) = Proj Span { w } ⊥ ( v ) = v- Proj w v = v- ( v,w ) ( w,w ) w = v- 4 4 w = v- w = 2- 1 3 - 1 1 1 1 = 1- 2 2- 1 . Alternate method: Gram-Schmidt. ker A has basis { 1- 1 , 1- 1 , 1- 1 } . Applying (some variant of) Gram-Schmidt yields the orthogonal basis { v 1 = 1- 1 ,v 2 = 1 1- 2 ,v 3 = 1 1 1- 3 } ....
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Math416_Midterm3_sol - Math 416 Abstract Linear Algebra...

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