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I am confused by what the question is askig in general. Thanks for any help

(a) Assume that the set {vector x, vector y, vector z} is linearly independent. Prove that the set

{5(vector x) + 2(vector y) + vector z,-2(vector x)-6(vector y), 4(vector x) + 3(vector y) + 8(vector z)}

is linearly independent.

(b) For any {vector x, vector y, vector z}, show that the set

{vector x + 2(vector y) + 3(vector z),-2(vector x) - (vector y) - 3(vector z), 4(vector x) + 3(vector y) + 7(vector z)} is linearly dependent by writing a dependence relation.

For a), define the matrix A = (x,y,z) where x,y and z are the linearly independent (row) vectors so that det(A) = 0. Let the coefficients multiplying the x,y,z vectors be formed into the matrix B so that the set of 3 vectors can be written (v1,v2,v3) = BA. We want to show that the vectors are linearly independent which involves showing that det BA ≠ 0. Using the property of determinants that det(BA) = (detB)(detA), and pluging in the coefficients toshow that det(B) ≠ 0 means det(BA) ≠ 0, so the vectors are linearly independent.

For b), using the same notation as in a), the coefficients in the matrix B give the result that det(B) = 0 so that, no matter whether the x,y,z, vectors are independent or not det(BA) = 0 and so the vectors are linearly dependent.

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