Let S = {v1, v2, v3} be a set of linearly independent vectors in R^3. Let S = {u

Question

Let S = {v1, v2, v3} be a set of linearly independent vectors in R^3.

Let S = {upsilon_1, upsilon_2, upsilon_3} be a set of linearly independent vectors in R^3. (a) Carefully explain what it means for the vectors in the set S to be linearly independent? (b) Are the vectors in the set L = {upsilon_1, upsilon_3} linearly independent? Show how you arrived at your answer. (c) Are the vectors in the set T = {upsilon_1 + upsilon_2, upsilon_1 - upsilon_3, upsilon_2 + upsilon_3} linearly independent? Show how you arrived at your answer.

Explanation / Answer

a)

The linear independence of the set S means the following: If c1v1+c2v2+c3v3 = 0 for some real numbers c1, c2, c3, then it must hold true that c1=c2=c3=0.

b)

The vectors in L are linearly independent, since if there exist scalars c1, c2 such that c1v1+c3v3=0, then it follows that  c1v1+0.v2+c3v3 = 0 , and hence by the linear independence of S, the set L (which is a proper subset of S) must also satisfy c1=c2=c3=0 (and c2 is already 0 in this case).

c)

Consider the identity 0 = (-1)(v1+v2)+(v1-v3)+(v2+v3). This shows that there exists a non-zero set of numbers c1=-1, c2=1,c3=1 that satisfies c1(v1+v2)+c2(v1-v3)+c3(v2+v3) = 0. Hence the vectors T are not linearly independent.

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