2114-Homework6 (1).pdf

2114-Homework6 (1).pdf

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Math 2114 Basis & Transformations Homework 6 Name: 1. Let A = - 1 2 3 2 - 1 2 0 2 - 1 2 2 2 6 1 0 have RREF B = 1 0 1 0 - 1 0 1 2 0 3 0 0 0 1 - 4 (a) Find a basis for Row A , call it B R = { ~ r 1 ,~ r 2 ,~ r 3 } (b) Find a basis for Null A , call it B N = { ~n 1 ,~n 2 } — Let B be the union of B R and B N so B = { ~ r 1 ,~ r 2 ,~ r 3 ,~n 1 ,~n 2 } . (c) Show that B is a basis for R 5 . — The next part is hard. I don’t expect you to have a full justification. I want to see what you think. Do your best. Full credit will be given for a good-effort attempt showing thought. (d) Will this always work? If you have a m × n matrix, with n > m , does combining a basis for Row A with a basis for Null A always produce a basis for R n ? To answer this you may want to consider the two-out-of-three step check for a basis for R n i. The set has n vectors. ii. The set is linearly independent. iii. The set spans R n 2. Let P 3 be the vector space of polynomials with degree at most 3. Let T be the transformation P 3 P 3 given by T : p ( x ) 7→ p ( - x ) + ( x + 1) · p 0 ( - x ) (a) Show that T is linear.

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