Suppose $U\subset W \subset V$ are three linear spaces with respective dimensions 3, 6 and 10. Let $E\subset L(V,V)$ be the set of linear transformations $f:V\rightarrow V$ such that $f(U)\subset U$ and $f(W)\subset W$.
Calculate $\dim\, E$.
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In general the linear transformation with a given square matrix $(a_{i,j})_{i,j=1}^n$ in a given basis $b_1,\ldots,b_n$ stabilises the subspace $\left<b_1,\ldots,b_d\right>$ if and only if $a_{i,j}=0$ whenever $j\leq d<i$. Since you can choose a basis such that $U,W,V$ are of this form (for $d=3,6,10$ respectively), your vector space is isomorphic, using this basis, to that of all matrices of the form $$ \begin{pmatrix} *&*&*&*&*&*&*&*&*&*\\*&*&*&*&*&*&*&*&*&*\\*&*&*&*&*&*&*&*&*&*\\ 0&0&0&*&*&*&*&*&*&*\\0&0&0&*&*&*&*&*&*&*\\0&0&0&*&*&*&*&*&*&*\\ 0&0&0&0&0&0&*&*&*&*\\0&0&0&0&0&0&*&*&*&*\\0&0&0&0&0&0&*&*&*&*\\0&0&0&0&0&0&*&*&*&*\\ \end{pmatrix} $$ where the $*$ designate arbitrary (and independent) values. The dimension is the number of times $*$ occurrs, which Frank Science counted correctly. |
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