invariant subspace

Let $T: V\rightarrow V$ be a linear transformation of a vector space $V$ . A subspace $U\subset V$ is called a $T$ -invariant subspace if $T(u)\in U$ for all $u\in U$ .

If $U$ is an invariant subspace, then the restriction of $T$ to $U$ gives a well defined linear transformation of $U$ . Furthermore, suppose that $V$ is $n$ -dimensional and that $v_1,\ldots, v_n$ is a basis of $V$ with the first $m$ vectors giving a basis of $U$ . Then, the representing matrix of the transformation $T$ relative to this basis takes the form $$ \begin{pmatrix} A & B \\ 0 & C \end{pmatrix}$$ where $A$ is an $m\times m$ matrix representing the restriction transformation $T\big|_U:U\to U$ relative to the basis $v_1,\ldots, v_m$ .