CHAPTER 02.20: VECTORS: Prove
that if a set of vectors is linearly dependent, then at least one vector can
be written as a linear combination of others In
this segment, we’ll talk about the proof to show that a set of vectors given
to you and what other null vectors, also called the zero vector. Can we show
that this set of vectors can be linearly dependent? So, let’s suppose
somebody says, let A1, A2, all the way up to Am, you are given m vectors be a
set of n dimensional vectors. So, your m vectors but there are dimension n.
So, let’s suppose although we’re not told exactly which one of them is zero -
let A1 be the null vector, again choose any of them to be the null vector.
So, if I choose A to be the null
vector, then, if I look at the linear combination k1 times A1 plus k2 times
A2 and I look at this linear combination and put that equal to zero vector. Now if I look at this linear
combination, then k1 not equal to 0, k2 equal to 0, and all them being 0, the
rest of them all being zero is a solution. So, this set of equations right
here - you’ll find out that if I choose k1 not equal to 0, if A1 is 0 so I
can multiply by any scalar and it is still a 0 vector. So, I can choose K1 to
be a non-zero number like 1, 2, 3 minus 5.3 whatever
I please. But the other ones I choose can be zero so what we are finding out
that this solution is non-trivial, not all zeros. Since the solution is non-trivial,
we have to say that A1, A2 all the way up to Am these m vectors linearly
dependent. That’s the end of this segment. |