Q2 (10 points) (a) Suppose that A is an invertible n x n matrik and that2,VER\"
ID: 3185357 • Letter: Q
Question
Q2 (10 points) (a) Suppose that A is an invertible n x n matrik and that2,VER" are linearly independent. Show that AVi,A Av are also linearly independent. (b) Find a square matrix, A, and a set of vectors. v that are linearly independent but for which Av, A, A are linearly dependent (c) Suppose that B is an invertible n! × m matrix and that wi, w2, , wk E Rm span R . Show that Bwi·BW2, ,Bi k also span R" (d) Find a square matrix, B, and a set of vectors wi.w2..w that span R" but for which Bw Bw.Bik do not span R".Explanation / Answer
Q2. (a). Let Av1, Av2,…,Avk be linearly dependent. Then, there exist scalars a1,a2,…,ak , not all 0, such that a1 Av1+a2 Av2+…+ ak Avk = 0, or, A(a1v1+a2 v2+…+ak vk) = 0. Now, on multiplying to the left by A-1, we have A-1A(a1v1+a2 v2+…+akvk)= A-1. 0 or, (A-1 A)(a1v1+a2 v2+…+akvk)=0 or, I (a1v1+a2 v2+…+ak vk) =0 or, a1v1+a2 v2 +…+ak vk = 0 which means that v1,v2,…,vk are linearly dependent, which is a contradiction. Hence, Av1, Av2,…,Avk are linearly independent.
(b). Let v1 = (1,0,0)T ,v2 = (0,1,0)T and v3 = (0,0,1)T. Then v1,v2,v3 are linearly independent. Also, let A =
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Then Av1 = (1,1,0)T,Av2 =(0,0,0)T and Av3 =(1,1,0)T . Now, since Av1 and Av3 are same, and since Av2 =(0,0,0)T, hence Av1,Av2,Av3 are linearly dependent.
( c). Let {Bw1, Bw2,…,Bwk} be a set which does not span Rm. Then there exists a vector v ? Rm, which cannot be expressed as a linear combination of Bw1, Bw2,…,Bwk. This implies that whatever be the value of the scalars a1,a2,…,ak, v ? a1Bw1+a2 Bw2+…+akBwk. Now, on multiplying to the left by B-1, we have B-1 v B-1 (a1 Bw1+a2 Bw2+…+ ak Bwk) or, B-1 v ? (B-1 B)( a1w1+a2w2+…+akwk) or, B-1 v ? a1w1+a2w2+…+akwk. This means that there is a vector B-1 v? Rm which cannot be expressed as a linear combination of w1,w2,…,wk i.e. the set { w1,w2,…,wk } does not span Rm , which is a contradiction. Hence, the set {Bw1, Bw2,…,Bwk} spans Rm.
(d). Let w1 = (1,0,0)T ,w2 = (0,1,0)T and w3 = (0,0,1)T. Then the set { w1,w2,w3} spans R3. Also, let B =
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Then Bw1 = (1,1,0)T,Bw2 =(0,0,0)T and Bw3 =(1,1,0)T . It is apparent that the set { Bw1, Bw2,w3}= {(1,1,0)T, (0,0,0)T } does not span R3.
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