help me on invariant subspace please

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.

please help me.

Quote:

Originally Posted by Kat-M

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.

Step 1: Let x be a nonzero vector in V. Then the one-dimensional subspace spanned by x is invariant under T. So $Tx = c_xx$ , for some scalar that (possibly) depends on x.

Step 2: Let x and y be linearly independent vectors in V. Then $Tx = c_xx,\ Ty = c_yy, \ T(x+y) = c_{x+y}(x+y)$ . Deduce from this that $c_x = c_y$ . Thus the constant c is in fact the same for every vector, and hence T is c times the identity.

thanks

thank you very much. you really helped me.