An algebra problem by Hobart Pao

Algebra Level 3

Given column vector \( A \in \mathbb{R}^n \) and row vector \( B \in \mathbb{R}^n\), if column vector A in rewritten as row vector B, such that entries \( A_{n1} \) in matrix A become entries \( B_{1n} \) in matrix B, where \( n \in \mathbb{N} \), then which of the following statements are necessarily correct?

  1. If vector A has a rank of 1, then vector B also has a rank of 1.

  2. If vector A is in reduced row-echelon form, then vector B will also be in reduced row-echelon form.

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