# Finding the basis with given transition matrix

$$P = \begin{bmatrix}1 & 1 & 0 \\ 0 & 1 & 3 \\ 3 & 0 & 1 \end{bmatrix}$$

a) P is the transition matrix from what basis B to the standard basis S = {e1, e2, e3} for R3?

b) P is the transition matrix from the standard basis S = {e1, e2, e3} to what basis B for R3?

My attempt:

For a), if PB=S (is this even right?), can we just multiply inverse of P both sides to get B?

• But the answers aren't right. In fact, there's a theorem here in the book that says that if A is any invertible matrix [ u1 | u2 | ..... | un ], then A is the transition matrix from the basis {u1 , u2 , ... , un} to { e1,e2,...,e3} – Zhi J Teoh Nov 11 '15 at 1:52

Your idea looks correct. Since S is the standard basis, i.e. S is the identity matrix, in a) your basis B are the columns of $P^{-1}S = P^{-1}$. In b) B equals P: $B = PS = P$.