Let $S$ be a spanning set for $V$ . Show that the vector $\mathbf{u}$ is in $V^{\perp}$ if and only if $\mathbf{u}$ is orthogonal to every vector in $S$ .

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Take $V$ a subspace of $\mathbb R^n$ and $S$ a spanning set for $V$ .

Take $\textbf{u}\in V^{\bot}$ then we have that $\textbf{u}\cdot\textbf{v}=0$ for every $\textbf{v}\in V$ . Since $S$ is a spanning set for $V$ we have that $S\subset V$ and then $\textbf{u}\cdot\textbf{v}=0$ for every $\textbf{v}\in S$ .

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