[Math] When $A^TA = I$, what matrix is orthogonal to what

Question

When $A^TA = I$, I am told it is orthogonal. What does that mean?

$A = \begin{bmatrix}cos\theta & & -sin\theta \\ \\ sin\theta & & cos\theta\end{bmatrix}, A^T = \begin{bmatrix}cos\theta & & sin\theta \\ \\ -sin\theta & & cos\theta\end{bmatrix}$

Answer 1

It means that the row vectors (and also the column vectors) form an orthogonal basis, that means if $A$ has dimension $n\times n$, $A$ consists of $n$ linear independent and pairwise orthogonal vectors spanning $\mathbb R_n$