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, letís some say somebody says ďHey, Iím going to give you three vectors a 1 equals 2, 25, 64, 144. Thatís 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, thatís 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 youíll 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 weíre adding them up; and you get that and from the next one youíll 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 thatís 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 itís 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, youíll trust the word for it; this is the only solution so since itís 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.