1. Let ? be a finite dimensional vector space with basis {?1,...,??} and ? ∈ L(?)....

1. 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 ?.

please also explain for (a)->(b) why are all the c coefficients 0 for all i>k? and why T(vk) in the span of (v1,.....,vk)? i need help understanding this.

Expert Solution

Colby Messinger answered 2 years ago

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