Find all invariant subspace of $A$ Consider a matrix mapping $A: V \to V$ for a vertor space V. Matrix $A$ has 3 eigenvalues are distinct : $\lambda_1,\lambda_2,\lambda_3$ and $v_1,v_2,v_3$ are vectors-corresponding .Find all invariant subspace of  $A$

The nontrivial invariant subspaces are just the eigenspaces, and sums of eigenspaces. Assuming this is a real vector space, these are
$$U_1 = \{cv_1 : c\in\mathbb R\}, U_2 = \{cv_2 : c\in\mathbb R\}, U_3=\{cv_3 : c\in\mathbb R\}.$$
Observe that if $u\in U_j$ then $u=cv_j$ for some $c\in\mathbb R$, hence $Au = A(cv_j)=cAv_j=c\lambda v_j\in U_j$. This means that $U_j$ is invariant under $A$.
However, $U_1+U_2$, $U_1+U_3$, and $U_2+U_3$ are also invariant subspaces, as if $u_i\in U_i$ and $u_j\in U_j$, $i\ne j$ then $u_i = c_iv_i$ and $u_j=c_jv_j$ for some $c_i,c_j\in\mathbb R$. So $$A(u_i + u_j)= Au_i + Au_j = c_iAv_i + c_jAv_j = c_i\lambda_iv_i + c_j\lambda_jv_j\in U_i+U_j.$$
