6) Show that vectors u1, u2 are linear independent if and only if u1 and u1 + u2

Question

6) Show that vectors u1, u2 are linear independent if and only if u1 and u1 + u2 are linear independent.

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Explanation / Answer

6) Suppose that the vectors u1 and u2 are linearly independent .

This means that any of their linear combination = 0 means the coefficients are zero.

ie., there exists scalars a, b such that au1 + bu2 = 0 implies a = b = 0.

Now, we have to prove that u1 and u1 + u2 are linearly independent.

Suppose for any two scalars c, d we have cu1 + d(u1 + u2) = 0 .ie., (c+d)u1 + du2 = 0.

But since by our assumption u1, u2 are linearly independent, we must have c+d =0 and d = 0

thus c = 0 as well

This means that cu1 + d(u1+u2) = 0 implies c = d = 0 .

Thus u1 and u1+ u2 are linearly independent

Conversely assume that u1and u1+u2 are linearly independent.

That is there exists scalars m, n such that mu1 + n(u1 + u2) = 0 implies m = n = 0.

But mu1 + n(u1+u2) = (m+n) u1 + nu2

Since m = 0, n = 0 , m + n is also 0.

Thus m+ n = 0 and n = 0, which are exactly the coefficients of u1 and u2.

Since m, n , u1, u2 are all arbitrary, it follows that u1 and u2 are linearly independent .

Hence the proof

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