Find the transition matrix $P_{S\leftarrow T}$

Question

Let $S=[1 \;1\;1],[1 \;2\;3],[1 \;0\;1]$ and $T=[0 \;1\;1],[1 \;0\;0],[1 \;0\;1]$. Find the transition matrix $P_{S\leftarrow T}$ from the set of ordered basis T to the set of ordered basis S.

All the examples in my text book are with column vectors as ordered basis and none with row vector; I cannot figure out how to tackle this problem. Any help is much appreciated.

Answer 1

The equality of column vectors $\mathbf{y}=\mathsf{A}\mathbf{x}$ holds iff the corresponding equality of row vectors $\mathbf{y}^{\mathsf{T}}=\mathbf{x}^{\mathsf{T}}\mathsf{A}^{\mathsf{T}}$ does (here $(\quad)^{\mathsf{T}}$ denotes taking the transpose). Consequently, if you're more comfortable with column vectors you can take the transpose of the original data, find the transition matrix, and transpose the result back.

Answer 2

Setting up the matrix like this $$ \left[ \begin{array}{ccc|c|c|c} 1&1&1&0&1&1\\ 1&2&3&1&0&0\\ 1&0&1&1&0&1 \end{array} \right] $$ If we reduce the left side to RREF, the right side will give us the transition matrix. Am I on the right track? Any help is much appreciated.