That should make sense.

So 1 times vector a plus 2 times vector b is equal to vector c.

Times the vector, let me just call this vector x, that right there is vector x.

So b plus a minus b is of course, going to be vector a.

A times the 0 vector is equal to the 0 vector.

I multiply this vector times this vector.

And then I have vector b.

And that's vector b.

My matrix a times vector x is equal to the 0 vector.

Maybe that's vector b right there.

That this is the same thing as the vector ac times x plus the vector bd times y, is equal to the vector ef.

Let's say I have the vector Let's call this vector a.

So it's the span of vector 1, vector 2, all the way to vector n.

the right linear combination this is called a linear combination.

The entire vector, and we know what that vector is.

Well, it's this vector plus that vector.

So it equals the span of that vector and that vector.

This is vector b, and this is vector a.

Let's say the vector a plus the vector v.

So if we just shift vector b over this way, well vector

So that's the length of the vector that's the magnitude of that vector

And what would the vector bd look like?

Now, we just did our first vector. We just normalized this one.

And then this vector right there would be the vector a minus b.

Plus x3 times the vector v3 plus the scalar x4 times the vector v4.

So a is equal to c1 times vector 1, plus c2 times vector 2, all the way to Cn times vector n.

So we could rewrite it vector 1, vector 2, vector 3, and vector 4.

And I want to know are there some combinations of vectors a and b where you can say, 5 times vector a, plus 3 times vector b, or 10 times victor a minus 6 times vector b some combination of vector a and b, where I can get vector c.

So this whole term right here is just this vector minus that vectors or it's this vector right there.

That is the sum of a and b.

I put the I put the head of the green vector to the tail of this magenta vector right over here. But the whole reason why did this is

Or that this is vector is being transformed into that vector.

And that also means that a times vector 2 is equal to our 0 vector.

We could also write I have my vector a, so I have my vector a.

Let's say that vector, let me call vector a, a1.

Maybe if I draw that vector, that vector might look something like this.

Inside vector

Vector c.

The vector ac.

This is equivalent to saying the span of the vector U1 and the vector V2, where U1 is the vector we got up here.

Notice if I take if I.. x starts at the tale of the green vector and goes all the way to the head of the magenta vector. vector and the magenta vector started the head of the green vector and then and then finishes I guess well where it finishes is where vector X finishes.

So this right here is going to be the length of vector v1, the length of this orange vector right here.

So v1 was equal to the vector ac, and v2 is equal to the vector bd.

So my question to you is, vector ef, we don't know where vector ef is.

So the transformation of vector a plus vector b, we could write it like this.