Krdytkyu

How do I explain that the rows of $A$ are unit vectors orthogonal to each other in $mathbb{R}^2$? I genuinely don't know where to start. Thanks in advance for any help.

Well, since A is a $2times 2$ matrix, one way to do this is brute force:

$$A=
begin{pmatrix}
a & b \
c & d \
end{pmatrix}$$

$$A^TA= begin{pmatrix}
a & b \
c & d \
end{pmatrix}
begin{pmatrix}
a & c \
b & d \
end{pmatrix} =
begin{pmatrix}
a^2 +b^2 & ac + bd \
ca +db & c^2 +d^2 \
end{pmatrix} = I_2$$

From there, you can conclude about $a^2+b^2$,$c^2+d^2$ and $ac+bd$

Writing the matrix as two row vectors $r_0,r_1$, the elements of the matrix $A^TA$ are

$$begin{pmatrix}r_0r_0^T&r_0r_1^T\r_1r_0^T&r_1r_1^Tend{pmatrix},$$ which are dot products. Conclude.

If $A=left(begin{smallmatrix}a&b\c&dend{smallmatrix}right)$, the first thing that you do is to compute $A^TA$; the second one is to see what it means to say that $A^TA=operatorname{Id}_2$. You will see that it means three things:

1. The first column has norm $1$;


2. The second column has norm $1$;


3. The columns ar orthogonal.