CHAPTER 02.15: VECTORS: Are these
vectors linearly independent? Example 1 In
this segment, we will take some examples to see whether we can show that set
of vectors in linear independent. So, let’s some say somebody says “Hey, I’m
going to give you three vectors a 1 equals 2, 25, 64, 144. That’s my first
vector and the second vector is given to you as 5, 8, and 12; and the third
vector is given to you as 1, 1, 1, and the question is asking you: are these
three vectors linearly independent? So, that’s the problem statement. So
if you take the linear combination of these three vectors what you are going
to get is k1 times 25, 64, 144, plus
k2 times 5 ,8, and 12, plus k3, 1, 1, 1, and you put that equals to 0 vector
which is 0, 0, 0. And then you can see you only get three questions three unknowns; so you’ll have let us put in
- write this down - so you would only get this equal to this 0, 0, 0, a
combination of 25 k1 plus 5 k2 and 1k3 here because 25 times k1 5 times k2
and 1 times k3 since we’re adding them
up; and you get that and from the next one you’ll get 64 k1 plus 8 k2 plus k3; and then here you
get 144 k1 plus 12 k2 plus k3 equal to
0. So basically what you have is three questions and three unknowns so this
is equal to 0, second question equal to 0, third question equal to 0. So
you have three questions and three unknowns and you only get the values of k1
k2 and k3 you can obviously see that k1 k2 k3 are there are 0 and still be a
solution in this set of questions, but that’s in fact the only solution. K1 equals to 0 K2 equals
to 0 k3 equals 0 is the only solution. And how do I know it’s the only solution?
That is something which you will learn in later chapters - how do we figure
that out that is the only solution. For right now, you’ll trust the word for
it; this is the only solution so since it’s the only solution that three; the
only solution and we have three vectors here that they are linearly
independent. That is the end of this segment. |