In any non-trivial vector space there will be several possible bases we could pick. In , for instance, we could use or . This first basis is known as the *standard basis*, and in general for an -dimensional vector space over , we’ll refer to as the standard basis, and will let denote the vector with a 1 in the -th position, and zeros elsewhere.

When we write down a vector like we implicitly mean that these are the coordinates to use with the vectors in the standard basis; these are the coefficients we multiply the basis vectors by to get describe our vector.

But how would we find the appropriate coordinates if we were to use as our basis? Let’s suppose our coefficients are . Then what we want to do is find the values of the such that

So what we have is a system of equations:

Notice that since the columns of this matrix form a basis for , the matrix is invertible, and so we can easily solve for the .

In general, if we have a basis , we will write

to mean that the are the coefficients of the vectors in our basis, and will let be the matrix converts the coordinates for the standard basis to coordinates in our basis. Likewise, will be the matrix which converts coordinates from the basis to coordinates with the standard basis.

Generalizing on the argument above we see that we can calculate by simply using our basis vectors as our columns. For we take the inverse of this matrix.

Now suppose we have two bases, and . The matrix which will take coordinates with respect to the basis and convert them into coordinates for the basis is denoted . One way to calculate is to convert our -coordinates into standard coordinates, and then convert those into -coordinates:

Alternatively, we could note that

(with the 1 in the -th spot)

(recall is the notation I use for the -th column of )

So the -th column of is simply -th vector from our basis, but in -coordinates.

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