help me on invariant subspace please

• Jan 7th 2009, 05:12 AM
Kat-M
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.

• Jan 7th 2009, 06:54 AM
Opalg
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 \$\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.
• Jan 7th 2009, 07:57 AM
Kat-M
thanks
thank you very much. you really helped me.