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What does it mean for a collection of bases $\{B_i\}_i$ for a corresponding set of vector subspaces $\{W_i\}_i$ to be pairwise disjoint? Thank you.

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Disjoint means "having empty intersection", so pairwise disjointness means $B_i \cap B_j = \emptyset$ for $i \ne j$.

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    @gary: Not necessarily: over $\mathbb{R}$, say $B_1=[(1,0), (0,1)]$ and $B_2 = [(0,-1),(1,0)]$. One of the change of bases matrix is \left(\begin{array}{cc}0&-1\\1&0\end{array}\right) which has no eigenvalues. More generally, take any invertible matrix that does not have $1$ as an eigenvalue, and use the columns to be one basis, and the standard basis to be the other basis. You seem to be assuming the change-of-basis will leave the common vector fixed, but it doesn't have to.2011-09-12