Prove that there is no isometry from $\mathbb R^3$ to $\mathbb R^2$ How should I start of? I know if $T$ is an isometry then $T=A \vec x+\vec b$ where A is orthogonal matrix and $\vec b$ is a column vector. If I can prove that there is no orthogonal matrix from $\mathbb{R^3}$ to $\mathbb{R^2}$ then will it suffice?(which I don't think is true?)
(Also my understanding of isometries is quite basic, I know isometries preserve distance and dot product)

Just prove that an isometry that fixes the origin preserves linear independence. And that $T$ is an isometry if and only if $A$ is. 
