3.8 Transition Matrices

Transition Matrix

Let V be a subspace of Rⁿ. Suppose S = {u1, u2, ..., uk} and T = {v1, v2,...vk} are bases for the subspace V. Define the transition matrix from T to S to be

P = ([v1]s [v2]s.... [vk]s)

The matrix whose columns are the coordinates of the vectors in T relative to the basis S.

P converts coordinates relative to the basis T to coordinates relative to the basis S.

The use of transition matrix

Let V be a subspace of Rⁿ. Suppose S = {u1,...,uk} and T = {v1, ..., vk} are bases for the subspace V. Let P be the transition matrix from T to S. Then for any vectors w in V

[w]s = P[w]T

Finding transition matrix

1) P = (S | T)

P = ( u1 u2 .... uk | v1 v2 .... vk)

2) rref: (Ik; 0(n-k)xk | P; 0(n-k)xk)

* Should focus on the first k rows of the rref, as there are k vectors in the basis.

When S = {u1, ..., uk} and T = {v1, ...., vk} be a basis for a subspace V in Rⁿ, P is the transition matrix from T to S.

Inverse of the transition matrix

given a transition matrix P from T to S, P-1 is the transition matrix from S to T.

*Cannot assume P is invertible though