please provide explanations within your proof
1) Prove that if ?1, ?2 are subspaces of ? such that ?1 + ?2 =?1 ? ?2 then there is a linear isomorphism
? : ?1 × ?2 ? ?1 ? ?2.
(Recall: Given two sets ?1, ?2 the cross product ?1 × ?2 isthe set of elements (?, ?)
where ? ? ?1 and ? ? ?2 with pointwise addition and scalarmultiplication.)
2) Let ? be a finite dimensional vector space with basis{?1,...,??} and ? ? L(?).
Show the following are equivalent:
(a) The matrix for ? is upper triangular. (b) ?(??) ?Span(?1,...,??) for all ?.
(c) Span(?1,...,??) is ?-invariant for all ?.
