Linear Algebra/OLD/Change of Basis

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Change of Basis[edit]

It was shown earlier that a square matrix can represent a linear transformation of a vector space into itself, and that this matrix is dependent on the basis chosen for the vector space. We will now show how to change the basis of a given vector space

Suppose we have some vector space whose basis is given by the set and we would like to change it to the set . The basis B still belongs to vector space aforementioned, so its vectors can be expressed as a linear combination [Eq. 1]

,

,

Each vector of the set B has a coordinate matrix with respect to the basis we started off with, namely the set designated as A. We represent this as

Setting these coordinate matrices as the columns of a matrix P gives us a transition matrix. This transition matrix transforms the original basis A to a new basis B of some vector space. The transition matrix is actually the transpose of [Eq. 1] that we saw earlier

To summarize, in order to find the transition matrix from some basis F to some basis G, we must compute the coordinate vector for each element of our original basis F with respect to the other basis G. The matrix whose columns are formed by the coordinate vectors is the transition matrix.

Proof[edit]

Theorem 1[edit]

If P is the transition matrix from the basis A to the basis Z, and β is an element of the vector space, then it follows that

Proof[edit]