3
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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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  • 7
    $\begingroup$ Suppose $\{e_1,e_2,e_3\}$ is a basis of $U$, and expand it to a basis of $W$, then to $V$, such as $\{e_1,e_2,e_3,f_1,f_2,f_3,g_1,g_2,g_3,g_4\}$. Consider the transformation matrix of $\varphi\in E$. I think the answer is $3\cdot3+3\cdot6+4\cdot10=67$. $\endgroup$ – Yai0Phah Jan 7 '13 at 11:34
  • $\begingroup$ I've edited your question. Please feel free to roll back if it doesn't reflect what you mean accurately. $\endgroup$ – user1551 Jan 7 '13 at 12:12
  • $\begingroup$ It's good, thanks! $\endgroup$ – Jonny Jan 7 '13 at 12:14
  • $\begingroup$ Frank Science's fine comment contains the answer. $\endgroup$ – Oliver Braun Jan 7 '13 at 12:23
  • $\begingroup$ @FrankScience please post as answer so I'll give You best answer $\endgroup$ – Jonny Jan 7 '13 at 12:24
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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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