An algebra problem by Hobart Pao

Algebra Level 4

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?

If vector A has a rank of 1, then vector B also has a rank of 1.
If vector A is in reduced row-echelon form, then vector B will also be in reduced row-echelon form.

Choice 2 only Choice 1 only Choices 1 and 2 Neither choice 1 nor choice 2

1 solution

Anand Raj
May 22, 2019

Since they are both row or column vectors, maximum rank can be 1. If the rank of A is 1, it just means that at least one of the element in A must be non zero. Thus correspondingly B's rank will also be 1 as it will also have at least one non zero element.

If A is in row echelon form, then the first element of A must be 1 and all the rest must be 0. Thus, B will also have first element as 1 in its first (and only) row, which satisfies the row echelon criteria. Therefore both statements are true.

An algebra problem by Hobart Pao

1 solution

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