How to write a linear combination of vectors

Solving systems of equations with matrices Video transcript In the last video we saw how a matrix and figuring out its inverse can be used to solve a system of equations.

How to write a linear combination of vectors

Linear dependence and independence Video transcript One term you are going to hear a lot of in these videos, and in linear algebra in general, is the idea of a linear combination. And all a linear combination of vectors are, they're just a linear combination. Let me show you what that means.

So let's say I have a couple of vectors, v1, v2, and it goes all the way to vn. And they're all in, you know, it can be in R2 or Rn. Let's say that they're all in Rn. They're in some dimension of real space, I guess you could call it, but the idea is fairly simple.

A linear combination of these vectors means you just add up the vectors. It's some combination of a sum of the vectors, so v1 plus v2 plus all the way to vn, but you scale them by arbitrary constants.

So you scale them by c1, c2, all the way to cn, where everything from c1 to cn are all a member of the real numbers.

how to write a linear combination of vectors

That's all a linear combination is. Let me show you a concrete example of linear combinations. Let me make the vector. Let me define the vector a to be equal to-- and these are all bolded. These purple, these are all bolded, just because those are vectors, but sometimes it's kind of onerous to keep bolding things.

So let's just say I define the vector a to be equal to 1, 2.

And I define the vector b to be equal to 0, 3. What is the linear combination of a and b? Well, it could be any constant times a plus any constant times b.

So it could be 0 times a plus-- well, it could be 0 times a plus 0 times b, which, of course, would be what? That would be 0 times 0, that would be 0, 0.

That would be the 0 vector, but this is a completely valid linear combination.

Linear Combination of Vectors

And we can denote the 0 vector by just a big bold 0 like that. I could do 3 times a. I'm just picking these numbers at random. So I'm going to do plus minus 2 times b. What is that equal to? Let's figure it out. Let me write it out. It's 3 minus 2 times 0, so minus 0, and it's 3 times 2 is 6.

This is a linear combination of a and b. I can keep putting in a bunch of random real numbers here and here, and I'll just get a bunch of different linear combinations of my vectors a and b. 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.

These are all just linear combinations. Now why do we just call them combinations? Why do you have to add that little linear prefix there?

Because we're just scaling them up. We're not multiplying the vectors times each other. We haven't even defined what it means to multiply a vector, and there's actually several ways to do it.

But, you know, we can't square a vector, and we haven't even defined what this means yet, but this would all of a sudden make it nonlinear in some form.

So all we're doing is we're adding the vectors, and we're just scaling them up by some scaling factor, so that's why it's called a linear combination. Now you might say, hey Sal, why are you even introducing this idea of a linear combination? Because I want to introduce the idea, and this is an idea that confounds most students when it's first taught.

Matrix word problem: vector combination (video) | Khan Academy

I think it's just the very nature that it's taught. Over here, I just kept putting different numbers for the weights, I guess we could call them, for c1 and c2 in this combination of a and b, right?Since your vectors are two-dimensional, any two of the vectors u, v and w should suffice to write b as a linear combination of them.

So let's arbitrarily choose u and v. We have. In fact, it is easy to see that the zero vector in R n is always a linear combination of any collection of vectors v 1, v 2,, v r from R n. The set of all linear combinations of a collection of vectors v 1, v 2,, v r from R n is called the span of { v 1, v 2,, v r }.

Write $\begin{pmatrix} 5 \\ 3 \\15 \end{pmatrix}$ as a linerar combination of the following vectors: $u=\begin{pmatrix} 1 \\ 2 \\5 \end{pmatrix}$, $v=\begin{pmatrix} 3 \\ -4 \\-1 \end{pmatrix}$, $w=\begin{pmatrix} -1 \\ 1 \\1 \end{pmatrix}$.

Writing a Vector as a Linear Combination of Other Vectors Sometimes you might be asked to write a vector as a linear combination of other vectors. This requires . Express a vector as a linear combination of given three vectors.

Midterm exam problem and solution of linear algebra (Math ) at the Ohio State University. Problems in Mathematics.

On the other hand, the constant function 3 is not a linear combination of f and g. To see this, suppose that 3 could be written as a linear combination of e it and e −it. This means that there would exist complex scalars a and b such that ae it + be −it = 3 for all real numbers t.

Linear combinations and span (video) | Khan Academy