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Linear Algebra 2270 Homework 8 preparation for the quiz on 07/15/2015 Problems: 1. Consider two different bases of R2 1 0 1 0 e = {[ ] , [ ]} , v = {[ ] , [ ]} 0 1 1 2 and two different bases of R3 : ⎡1⎤ ⎡0⎤ ⎡0⎤⎫ ⎡1⎤ ⎡0⎤ ⎡0⎤⎫ ⎧ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎢⎢ ⎥⎥ ⎢⎢ ⎥⎥ ⎢⎢ ⎥⎥⎪ ⎪ ⎪⎢⎢ ⎥⎥ ⎢⎢ ⎥⎥ ⎢⎢ ⎥⎥⎪ ⎪ e = ⎨⎢0⎥ , ⎢1⎥ , ⎢0⎥⎬ , w = ⎨⎢1⎥ , ⎢2⎥ , ⎢1⎥⎬ ⎥ ⎥ ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ ⎢ ⎢ ⎢ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎢0⎥ ⎢0⎥ ⎢1⎥⎪ ⎪ ⎪ ⎪ ⎩⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎪ ⎭ ⎩⎢⎣0⎥⎦ ⎢⎣0⎥⎦ ⎢⎣1⎥⎦⎪ ⎭ ′ (a) Let T ∶ R3 → R2 ⎡ ⎤ ⎛⎢⎢x1 ⎥⎥⎞ x + 2x2 + x3 ] T ⎜⎢x2 ⎥⎟ = [ 1 ⎥ ⎢ x1 − x3 ⎝⎢x3 ⎥⎠ ⎣ ⎦ Find the following matrices of T in different bases: Me′ ,e (T ), Mw,e (T ), Me′ ,v (T ), Mw,v (T ) (b) Let T ∶ R3 → R3 ⎡ ⎤ ⎡ ⎤ ⎛⎢⎢x1 ⎥⎥⎞ ⎢⎢x1 + 2x2 + x3 ⎥⎥ T ⎜⎢x2 ⎥⎟ = ⎢ x1 − x3 ⎥ ⎥ ⎝⎢⎢x3 ⎥⎥⎠ ⎢⎢ ⎥ x2 ⎣ ⎦ ⎣ ⎦ Find the following matrices of T in different bases: Me′ ,e′ (T ), Mw,e′ (T ), Me′ ,w (T ), Mw,w (T ) (c) A matrix I ∈ Rn×n is called an identity matrix, if where it has 1’s. ⎡1 0 0 ⎢ ⎢0 1 0 ⎢ ⎢ I = ⎢0 0 1 ⎢ ⎢⋮ ⋱ ⎢ ⎢0 0 0 ⎣ it has 0’s everywhere apart of the diagonal, . . . 0⎤⎥ . . . 0⎥⎥ ⎥ . . . 0⎥ ⎥ 0⎥⎥ . . . 1⎥⎦ Consider the identity transformation T ∶ R3 → R3 ⎡ ⎤ ⎡ ⎤ ⎛⎢⎢x1 ⎥⎥⎞ ⎢⎢x1 ⎥⎥ T ⎜⎢x2 ⎥⎟ = ⎢x2 ⎥ ⎝⎢⎢x3 ⎥⎥⎠ ⎢⎢x3 ⎥⎥ ⎣ ⎦ ⎣ ⎦ Find the following matrices of T in different bases: Me′ ,e′ (T ), Mw,e′ (T ), Me′ ,w (T ), Mw,w (T ). Which of those are 3 × 3 identity matrices? 2. Consider a set of matrices Rm×n with the operation of addition and scalar multiplication, which is the same as for vectors, but the numbers are arranged in a table, not in a column: for any c ∈ R: ⎡ a1 1 . . . a1 n ⎤ ⎡ b1 1 . . . b1 n ⎤ ⎡ a1 1 + b1 1 . . . a1 n + b1 n ⎤ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⋮ ⎥ ⋱ ⋮ ⎥+⎢ ⋮ ⋱ ⋮ ⎥=⎢ ⋮ ⋱ ⋮ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢am 1 . . . am n ⎥ ⎢bm 1 . . . bm n ⎥ ⎢am 1 + bm 1 . . . am n + bm n ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (1) ⎡ a1 1 . . . a1 n ⎤ ⎡ c⋅a1 1 . . . c ⋅ a1 n ⎤ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎥ ⋱ ⋮ ⎥=⎢ ⋮ ⋱ ⋮ ⎥ c⋅⎢ ⋮ ⎢ ⎥ ⎢ ⎥ ⎢am 1 . . . am n ⎥ ⎢c ⋅ am 1 . . . c ⋅ am n ⎥ ⎣ ⎦ ⎣ ⎦ (2) (a) Prove that “⋅” and “+” satisfy V1) – V8), and as a result Rm×n is a vector space. (Hint: the proof is the same as for vectors, just the numbers are arranged in a table, not in a column) 1 (b) What is the dimension of R2×3 ? (c) Prove that the set of 2 × 2, symmetric matrices: a a W = {[ 1 1 1 2 ] ∶ a2 1 = a1 2 } a2 1 a2 2 is a subspace of R2×2 . (d) Prove that the set of 3 × 3 upper triangular matrices: ⎡a ⎧⎡⎢a1 1 a1 2 a1 3 ⎤⎥⎫ ⎫ ⎧ a a ⎤ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎢ ⎪ ⎪ ⎪⎢⎢ 1 1 1 2 1 3 ⎥⎥ ⎥⎪ 3×3 ∶ a2 1 = a3 1 = a3 2 = 0⎬ = ⎨⎢ 0 a2 2 a2 3 ⎥⎬ W = ⎨⎢a2 1 a2 2 a2 3 ⎥ ∈ R ⎥ ⎢ ⎥ ⎢ ⎪ ⎪ ⎪ ⎪ ⎪⎢ 0 ⎪ ⎪⎢a ⎪ ⎪ ⎪ 0 a3 3 ⎥⎦⎪ ⎭ ⎩⎣ ⎭ ⎪ ⎩⎣ 3 1 a3 2 a3 3 ⎥⎦ is a subspace of R3×3 . 2