On proving Linearly independence

1. Feb 3, 2004

franz32

Hello. I want to ask questions... I hope you can guide me
in showing the proof.

1. Let a, b and c be vectors in a vector space such that {a, b} is linearly independent. Show that if c does not belong to span {a, b}
, then {a, b, c} is linearly independent.

I know that is {a,b} is l. independent, it implies that
c1a + c2b = 0. That is, c1 = c2 = 0.
What does it mean (imply) when c is not in span {a,b}?
How will I show the essence of the proof?

2. Let S = {u1, u2, ..., uk) be a set of vectors in a vector space, and let T = {v1, v2, ..., vm}, where each vi, i = 1, 2, ..., m, is a linear combination of the vectors in S. Show that

w = b1v1 + b2v2 + ... + bmvm
is a linear combination of the vectors in S.

How will I show the essence of the proof? I don't understand the meaning (implication)of the first sentence.

Last edited: Feb 3, 2004