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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.

nova_star
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2 Answers2

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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.

K B Dave
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  • @K B Dave, I tried setting up the matrix without taking transpose. But I got your point about using the transpose too. – nova_star Mar 05 '18 at 18:50
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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.

nova_star
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