Writing a vector as a linear combination
I could never-- there's no combination of a and b that I could represent this vector, that I could represent vector c.
I can find this vector with a linear combination. So let's say that my combination, I say c1 times a plus c2 times b has to be equal to my vector x.
If I had a third vector here, if I had vector c, and maybe that was just, you know, 7, 2, then I could add that to the mix and I could throw in plus 8 times vector c.
And that's why I was like, wait, this is looking strange. That's going to be a future video.
Express the vector as a linear combination of i and j
So x1 is 2. Over here, when I had 3c2 is equal to x2 minus 2x1, I got rid of this 2 over here. So that one just gets us there. And all a linear combination of vectors are, they're just a linear combination. So let's say that my combination, I say c1 times a plus c2 times b has to be equal to my vector x. Now why do we just call them combinations? Well, I can scale a up and down, so I can scale a up and down to get anywhere on this line, and then I can add b anywhere to it, and b is essentially going in the same direction. This is for this particular a and b, not for the a and b-- for this blue a and this yellow b, the span here is just this line. About Linear Combinations of Vectors — The Basics In linear algebra, we define the concept of linear combinations in terms of vectors. So if I were to write the span of a set of vectors, v1, v2, all the way to vn, that just means the set of all of the vectors, where I have c1 times v1 plus c2 times v2 all the way to cn-- let me scroll over-- all the way to cn vn. This is just 0. The span of the vectors a and b-- so let me write that down-- it equals R2 or it equals all the vectors in R2, which is, you know, it's all the tuples. You get let me write it in a different color.
Why is this true? I mean, if I say that, you know, in my first example, I showed you those two vectors span, or a and b spans R2.
Which of the following are linear combinations of
And there's no reason why we can't pick an arbitrary a that can fill in any of these gaps. But it begs the question: what is the set of all of the vectors I could have created? Well, a linear combination of these vectors would be any combination of them using addition and scalar multiplication. Let's ignore c for a little bit. Let me draw it in a better color. That would be the 0 vector, but this is a completely valid linear combination. So we can fill up any point in R2 with the combinations of a and b. These form a basis for R2. So let's see if I can set that to be true. Well, I can scale a up and down, so I can scale a up and down to get anywhere on this line, and then I can add b anywhere to it, and b is essentially going in the same direction.
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