Syndrome is Zero iff Vector is Codeword/Mistake

Source Work
Chapter $6$: Error-correcting codes
 * Proposition $6.20$: Proof

Mistake

 * Hence, the element $c$ is a codeword  $c = u G = u \paren {\begin{array} {c|c} I & A \end{array} } = \paren {\begin{array} {c|c} u & v \end{array} }$, where $v = u A$, and $\paren {\begin{array} {c|c} U & v \end{array} }$ denotes the concatenation of $u$ with $v$, ...''

Correction
Where it says:
 * where ... $\paren {\begin{array} {c|c} U & v \end{array} }$ denotes the concatenation of $u$ with $v$

it should have:
 * $\paren {\begin{array} {c|c} u & v \end{array} }$