User:Alexathkust/temp

theorem 7.3
Let T be a linear operator on a finite-dimensional vector space V such that the characteristic polynomial of T splits, and let $$\lambda_1,\lambda_2,...,\lambda_k$$ be the distinct eigenvalues of T. Then, for every $$x\in \rm{V}$$, there exist vectors $$v_i\in \rm{K}_{\lambda_i}, 1\le i\le k$$, such that
 * $$x=v_1+v_2+\cdots+v_k$$

Proof $$m$$: multiplicity of $$\lambda_1$$ or $$\lambda_i$$ $$f(t)$$: characteristic polynomial of T.

mathematic induction.
 * 1.)k=1,(T-&lambda;I)m=0

counter example
differentiable but partial derivative not continuous: http://www.math.umn.edu/~rogness/mathlets/partialsNotContDiff.html

not continuous at (0,0) http://www.math.tamu.edu/~tvogel/gallery/node14.html