Examples of using Basis vectors in English and their translations into Polish
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So it's the coefficient on the basis vectors.
Those are out basis vectors, so our third column is going to be 0, 0, minus 1.
Well, vector 2 is one of the basis vectors.
In general, the basis vectors are neither unit vectors
these are the basis vectors, right?
Well all of these guys are basis vectors so they're all going to be associated with pivot vectors or they're all going to be associated with pivot columns.
all the way through vk, these are the basis vectors.
The basis for V is-- or V is spanned by these basis vectors, which is the columns of these.
those would make for good basis vectors.
We could try to apply the transformation to our standard basis vectors in R3 like we have done in the past.
I have talked a lot about the idea that eigenvectors could make for good bases or good basis vectors.
and v3, because these basis vectors, it's very easy to figure out what their transformations are.
You know, the transformation matrix in the alternate basis-- this is one of the basis vectors.
because you have k basis vectors in your basis, or you have k vectors in your basis. .
then I can just apply that to my basis vectors.
And then if we had more basis vectors, we would say plus V2 dotted with our next basis vector, times that basis vector, and so on and so forth.
Where A is equal to essentially, or exactly, the matrix with the basis vectors as columns.
then apply those coefficients times the basis vectors, add them up and you know your projection.
it's equal to these ci's times their respective basis vectors, but now we know what the ci's are.
Now when I take the dot product of one of my basis vectors, the ith basis vector,
because that's the coefficients on our basis vectors.
Their direction or the lines they span fundamentally don't change. And the reason why they're interesting for us is, well, one of the reasons why they're interesting for us is that they make for interesting basis vectors-- basis vectors whose transformation matrices are maybe computationally more simpler, or ones that make for better coordinate systems.
Let's say I set up some matrix A, that has my basis vectors as the columns-- So if I set up some matrix A that looks like this, v1, v2,
They're that basis vector times your vector x.
It only has the basis vector U1 right here.
We say ci is just the ith basis vector dotted with x.
And there's our second basis vector.
That's its first basis vector.
But V1 only has one basis vector.
This first basis vector right there. v2 is the second one, and then you go