So I'm currently doing an advanced linear algebra module and I'm confused on two statements from two different lectures. I'll attach a section of two lectures and highlight the statements that confuses me. (I apologize for poorly typing the equations, I don't know yet how to type formulas on this site)
My understanding is that $\langle v_i, v_j \rangle = \langle v_j, v_i \rangle$ due to symmetry, but in the last photo it states that $\langle v_i, v_j \rangle = 0$ if $i<j$. This means that $\langle v_j, v_i \rangle$ might not be 0. Why is that? Why does the symmetry axiom not hold? I think I'm missing something. Thanks in advance!
First lecture, discussing what is an inner product and its axioms / 4th lecture, discussing QR decomposition
