Re: Change of co-ordinates

presumably, you mean , that is, . suppose we have a vector in ginve in C-coordinates, that is:

or . in other words,

.

now if we had such a matrix A. then:

would be its i-th column. but that would just be the i-th basis vector of C in B-coordinates. so, for example,

, since:

convince yourself that a linear combination of the basis vectors in C is taken to the same linear combination of the images of the basis vectors in B-coordinates, because a matrix is a linear mapping.

Re: Change of co-ordinates

Another approach: by a well known theorem if are basis of a vector space and then, where

In our case, we immediately get

Re: Change of co-ordinates

Quote:

Originally Posted by

**Deveno** presumably, you mean

, that is,

. suppose we have a vector in

ginve in C-coordinates, that is:

or

. in other words,

.

now if we had such a matrix A. then:

would be its i-th column. but that would just be the i-th basis vector of C in B-coordinates. so, for example,

, since:

convince yourself that a linear combination of the basis vectors in C is taken to the same linear combination of the images of the basis vectors in B-coordinates, because a matrix is a linear mapping.

Yes , and

so can we say that M=A (inverse), where

M=3 7 1

2 5 1

7 4 -9

and just find the inverse of M to find our matrix A?

Re: Change of co-ordinates

no, the matrix:

is the matrix A, it changes C-coordinates into B-coordinates.

if you have a non-standard basis, and you want to change coordinates from the non-standard basis to the standard basis,

you write down the matrix whose columns are the non-standard basis vectors coordinates in the standard basis,

that is the "change of coordinate" matrix. if you want to change from B-coordinates to C-coordinates, THEN you would take the inverse of A.