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, weíll 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, letís 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, letís suppose although weíre 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 zerovector. 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 - youíll 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. Thatís the end of this segment.