Let A € Mnxn (F). Prove the following statements. (a) If {v₁,..., Un} is a basis for F" that triangularizes LA, then the matrix Q = (v₁ Un) € Mnxn (F) is invertible, and Q¹AQ is upper triangular. (b) Conversely, if Q € Mnxn (F) is an invertible matrix such that Q-¹AQ is upper triangular, then the column vectors of Q form a basis for F" that triangularizes LA.

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