CHAPTER 02.13: VECTORS: What do you mean by linear combination of vectors? Example
In this segment, we’ll talk about an example of a linear combination of vectors. Let’s suppose that somebody give you – says that “Hey, you have an A vector which is given to you as 2, 3, 6, and then B vector is given as 5, -2, 3” And they give another vector C, let’s suppose, and that’s given as 8, 1, 2. And somebody says “Hey, can you find a linear combination of these vectors which is 5 times A plus 2 times B plus 3 times C.” So find this. So all you’re doing is you’re taking a vector A multiplying by 5; vector B multiplying by 2; vector C multiplying by 3. And then you want to add them all together to be able to find the linear combination. So all I did is 5 times A vector, which is 2, 3, and 6. Then 2 times the B vector which is 5, -2, 3 plus 3 times the C vector, which is 8, 1, and 2. So, as we have mentioned that when we multiply vector by a scalar, each component gets multiplied by that quantity. So the first five times A becomes 10, 15, 30. The second vector which is 2 times B becomes 10, -4, and 6. And the third vector which is 3 times C, here; each component is modified by 3, becomes 24, 3, and 6. And now what we’re going to do is we have to add each of the components correspondingly. So first element here, first element here, first element here, and so on and so forth. So first element here 10 plus 10 plus 24 is 44. 15 minus 4 is 11; 11 plus 3 is 14; 30 plus 6 is 36, plus 6 is 42. So that is your linear combination of those three vectors. And that’s the end of this segment.