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 we’ll 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,
I’ll get zero vector equal to zero vector so there is that trivial solution, but
if that’s 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 k’s are equal to zero, then it’s 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 it’s considered
linearly dependent. And that’s the end of this segment. |