I have some questions about linear algebra. Let's say $\{|v_1 \rangle, |v_2 \rangle, |v_3 \rangle \}$ are orthonormal basis of the $\mathcal{V}(\mathbb{C})$. Then, let's define two vectors $$ |a \rangle = 2i |v_1 \rangle - |v_ \rangle + 4 |v_3 \rangle \\ |b \rangle = |v_1 \rangle +3i |v_2 \rangle - |v_3 \rangle $$
My questions are:
1. Is it correct?
$$ \langle a|b \rangle = -2i \langle v_1|v_1 \rangle - 3i \langle v_2|v_2\rangle - 4 \langle v_3|v_3 \rangle = -2i - 3i -4 = -4 - 5i $$
or should there be some more expression above?
2. Does product of two kets $ |a\rangle|b\rangle$ make sens?
In my opinion no, because kets are columns and product of columns like that's is indefinite. But maybe I'm wrong and there is better explanation?
3. Does sum $ |\rangle + \langle| $ make sense?
Same as above. Bra is a row and ket is a column, so it's impossible to sum them.
Can somebody tell me if I'm wrong or explain 2. and 3. more mathematically?