Let T Rnright arrowRm be a linear transformation and let v1

Let T: Rnright arrowRm be a linear transformation, and let {v1, v2, v3} be a linearly dependent set in Rn. Explain why the set (T(vi), T(v2), T(v3)} is linearly dependent. Choose the correct answer below. A. Given that the set {v1, v2, v3} is linearly dependent, there exist c1, c2, c3, not all zero, such that c1v1 + c2v2 + c3v3 = 0. It follows that C1T(v1) + c2T(v2) + c3T(v3) not equal to 0. Therefore, the set T(v1), T(v2), T(v3)} is linearly dependent. B. Given that the set {v1, v2, v3} is linearly dependent, there exist c1, c2, c3, all zero, such that c1v1 + c2v2 + c3v3 = 0. It follows that c1T(v1)+c2T(v2)+ c3T(v3) = 0. Therefore, the set T(v1), T(v2), T(v3)} is linearly dependent. C. Given that the set {v1, v2, v3} is linearly dependent, there exist c1, c2, c3, not all zero, such that c1v1 + c2v2 + c3v3 = 0. It follows that CiT(v1) + c2T(v2) + c3T(v3) = 0. Therefore, the set T(v1), T(v2), T(v3)} is linearly dependent. D. Given that the set {v1, v2, v3} is linearly dependent, there exist c1, c2, c3, not all zero, such that c1v1 + c2v2 + c3v3 not equal to 0. It follows that C1T(v1) + c2T(v2) + c3T(v3) not equal to 0. Therefore, the set T(v1), T(v2), T(v3)} is linearly dependent.

Solution

Option A is correct answer.

There exist C1, C2, C3 not all zero such that C1V1+C2V2+C3V3=0. That means V1, V2, V3 can all be written as the combination of other two vectors. Therefore they are linearly dependent.

And since C1, C2, C3 are not all 0 , C1T1+C2T2+C3T3 can not be equal to 0 unifromly..Therefore T1 T2 T3 are linearly dependent.