Prove that every orthogonal matrix has a null space {0}

Question

I am currently working on some practice problems regarding orthogonality and its properties, and one of the proofs I am trying requires that I show prove that "every orthogonal matrix has a null space {0}". The solution provided by my professor only states: $Q^{-1} = Q^{T}$ with no explanation. If anyone could offer some guidance on how to go about writing a more detailed proof the statement, I would greatly appreciate it! Thank you!

Answer 1

Note that for an orthogonal matrix $Q$, we have $Q^{-1}=Q^T$. This means $Q$ is invertible (its inverse is $Q^T$), and we know that invertible matrices have trivial kernels (i.e. their null space is $\{0\}$).

Answer 2

Let $A\in M_{n \times m}(\mathbb{K})$ for some field $\mathbb{K}$. Since $A$ is orthogonal then by definition $A^{t}=A^{-1}$ it implies that $A$ is a invertible matrix.

Now consider the null space given by $$Ker(A)=\lbrace x\in \mathbb{R^m} \mid A \cdot x=0_{n \times m}\rbrace$$ $$ Ker(A)=\lbrace x\in \mathbb{R^m} \mid x=0_{m \times n}A^{-1}\rbrace $$ $$Ker(A)=\lbrace x=(0,0,0, \cdots,0)\rbrace$$ $$Ker(A)=\lbrace 0 \rbrace$$