CHAPTER 02.14: VECTORS: What is a vector?



In this segment, we will talk about what does it mean by vectors being linearly independent. So, let A1, A2, all the way up to Am be vectors of n-dimension. Let k1, k2, all the way up to km be scalars; then if k1 times A1 - so what we are doing is we are taking a linear combination of the m vectors. What we will do to this linear combination of m vectors where k1, k2, km are just scalars and weíll put them equal to the zero vector; has the only solution as k1=0, k2=0 all the way up to km=0. Then A1, A2, all the way up to the Am vector which is given to you are linearly independent. It seems obvious that if I put k1, k2, all the way up to km equal to zero, Iíll get zero vector equal to zero vector so there is that trivial solution, but if thatís the only solution then it is linearly independent. But if you find out that hey - there is some other solution other than the trivial solution with this one where k1 and all the kís are equal to zero, then itís not considered linearly independent then its linearly dependent so the only solution is the previous solution when its linearly independent if you have more than one solution other than the previous solution then itís considered linearly dependent. And thatís the end of this segment.