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369 HW5 problems 1. Let 1 0 2 β = 2 , 0 , 1 , 2 1 0 4 v = 3 . 4 Show that v is in the span of β. Why doesn’t it make sense to write coordinates of v with respect to β? 2. Let L : R2 → R3 be defined by x + 2y x L = 2x − 3y , y y and let v= 2 , −3 1 0 D= , , 1 1 0 0 1 R = 1 , 1 , 0 . 0 1 1 Compute L(v) and [L(v)]R . Then compute [L]RD and [v]D , and check that [L(v)]R = [L]RD [v]D . 3. Let 1 A = 0 3 2 −1 2 −4 1 , 10 0 0 1 β = 0 , 1 , 0 . 1 1 1 Show that β is a basis for R3 . Compute [A]std β , [A]β std and [A]ββ . 4. With β and A as in Problem 3, compute also [I]β, std and [I]std, β . Check that [I]−1 β, std = [I]std, β . Then check that A = [I]std, β [A]β,β [I]β, std . 5. Let A= 1 2 2 . 1 Find a basis β = {v1 , v2 } for R2 such that [A]ββ is diagonal. Then find P so that A = P CP −1 , where C := [A]ββ . 6. Let A= 1 0 2 , 1 B= 1 1 2 . 2 (a) Is it possible to find a basis β = {v1 , v2 } for R2 such that v1 , v2 have the property Avj = λj vj for some scalars λj , j = 1, 2? Why or why not? (b) Find a basis β = {v1 , v2 } for R2 such that v1 , v2 have the property Bvj = λj vj for some scalars λj , j = 1, 2. Use this to write down [B]ββ . What do the diagonal entries of [B]ββ tell you about the dimension of the image of B? 1