I'm looking at the notes on https://www.cs.cornell.edu/~bindel/class/cs3220-s12/notes/lec11.pdf.
On the first page, we have the following steps
\begin{align} ||Ax-b||^2&=||Q^T(Ax-b)||^2\\ &=\left|\left|\begin{bmatrix}R_{11}\\0\end{bmatrix}x-\begin{bmatrix}Q_1^Tb\\Q_2^Tb\end{bmatrix}\right|\right|^2\\ &=||R_{11}x-Q_1^Tb||^2+||Q_2^Tb||^2 \end{align}
How did the 3rd line follow from the 2nd? I don't understand how the single norm was broken into the addition of 2 norms.
If $u \in \mathbb R^n$ and $v \in \mathbb R^m$ are column vectors then $$ \left\| \begin{bmatrix} u \\ v \end{bmatrix} \right \|^2 = \| u \|^2 + \| v \|^2. $$ The reason is that both are equal to $u_1^2 + \cdots + u_n^2 + v_1^2 + \cdots + v_m^2$.
So in this example we are saying that $$ \left \| \begin{bmatrix} R_{11} x - Q_1^T b \\ Q_2^T b \end{bmatrix} \right \|^2 = \| R_{11}x - Q_1^T b \|^2 + \| Q_2^T b \|^2. $$