let T denote a linear operator on a vector space V. suppose that every subspace of V is invariant under T. prove that T is a scalar multiple of the identity map.
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Step 1: Let x be a nonzero vector in V. Then the one-dimensional subspace spanned by x is invariant under T. So $\displaystyle Tx = c_xx$, for some scalar that (possibly) depends on x.
Step 2: Let x and y be linearly independent vectors in V. Then $\displaystyle Tx = c_xx,\ Ty = c_yy, \ T(x+y) = c_{x+y}(x+y)$. Deduce from this that $\displaystyle c_x = c_y$. Thus the constant c is in fact the same for every vector, and hence T is c times the identity.