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# linear transformation & change of basis watch

1. Hi I got a little confused on the statements below:

1) P is the transition matrix from B coordinates to standard coordinates.
2) P^-1 is the transition matrix from standard coordinates to coordinates in the basis B.
3) P is ALSO the matrix of the linear transformation T:e1 -> v1

e= standard basis vector
v=new basis vector

My question is since P is from B coordinates to standard, why is it that during linear transformation, it seems to me that it's mapping standard back to basis (the other way round).

I'm genuinely confused and would appreciate if someone could kindly answer this (although the question may be considered silly to some geniuses out there)
2. (Original post by blahlostape)

I'm genuinely confused and would appreciate if someone could kindly answer this (although the question may be considered silly to some geniuses out there)
Yes, I usually have to think about this several times, probably 'cause I don't use it much.

In co-ordinate form, consider the vector (1,0,0,...,0) if you feed this to P, then it gives you the vector v1 in terms of the standard basis.

So, although P converts from the B basis to the standard basis, if you feed in the standard basis (relative to the standard basis) you will get the B basis (relative to the standard basis)

Clear as mud?
3. (Original post by ghostwalker)
Yes, I usually have to think about this several times, probably 'cause I don't use it much.

In co-ordinate form, consider the vector (1,0,0,...,0) if you feed this to P, then it gives you the vector v1 in terms of the standard basis.

So, although P converts from the B basis to the standard basis, if you feed in the standard basis (relative to the standard basis) you will get the B basis (relative to the standard basis)

Clear as mud?
4. I think your confusion comes from the fact that vectors doesn't necessarily mean column vectors.

If B = {b1, b2, b3,..., bn} is a basis where bi are vectors, these vectors could be matrices, or different column vectors than the standard basis for R^n.

You use a column vector to represent a linear combination of the basis vectors. I.e. (1 2 3 ...) represents the vector v=b1 + 2b2 + 3b3 + ...

In the case of R^n we use column vectors to represent column vectors I.e. in R^2:

(1 1) represents 1 x (1 0) + 1 x (0 1)=(1 1) (this is a point in space, the other represents a combination of vectors) in the standard basis.
In the basis B={(1 1), (1 2)}, (1 1) represents 1 x (1 1) + 1 x (1 2)=(2 3) which is a completely different point in space.

Imagine my row vectors as columns,I don't know how to do column vectors in latex .

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