Question: Let L: V V be a linear operator. A nonempty sub-space U of V is called invariant under L if L(U) is contained in U.

Let L: V †’ V be a linear operator. A nonempty sub-space U of V is called invariant under L if L(U) is contained in U. Let L be a linear operator with invariant subspace U. Show that if dim U = m and dim V = n, then L has a representation with respect to a basis S for V of the form

Where A is m x m, B is m x (n - m), O is the zero (n - m) x m matrix, and C is (n - m) x (n - m).

B C

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