please provide explanations within your proof 1) Prove that if ?1, ?2 are subspaces of ?...

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 is the set of elements (?, ?)

where ? ∈ ?1 and ? ∈ ?2 with pointwise addition and scalar multiplication.)

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 ?.

Expert Solution

Colby Messinger answered 2 years ago

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