Exercise 2.7.7

Question

Prove that if $A \mathbf{x} = \mathbf{0}$ has unique solution, then the equation $A^{\mathsf{T}} \mathbf{x} = \mathbf{b}$ has a solution for every right side $\mathbf{b}$. (	extit{Hint:} count pivots)

Jing Huang · Accepted Answer

\begin{proof}
Suppose $A \in \mathbb{R}^m 	imes \mathbb{R}^n$. Note that for $A \mathbf{x} = \mathbf{0}$, there is always a trivial solution $\mathbf{x = 0} \in \mathbb{R}^n$. And we know the trivial solution is unique, which also indicates that the echelon form of $A$ has a pivot at every column. Accordingly, the echelon form of $A^{\mathsf{T}}$ has a pivot at every row (Think that the echelon form of $A^{\mathsf{T}}$ is completed by column reduction that corresponds to the row reduction of $A$). So $A \mathbf{x = b}$ is consistent for any $\mathbf{b}$.   
\end{proof}

Exercise 2.7.7

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Exercise 2.7.7

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