A computational question in an example in Section-8.2 of Inner Product Spaces

Question

I have a question in section- 8.2 of Ch-Inner Product Spaces of Hoffman Kunze Linear Algebra.

It's a computational question in an example( Example 13 , page 283) :

The question is in 2nd line of proof: how does ((c, d) |(a, b)) lead expression in 3rd line which is (c × conjugate of a + d×conjujate of b) ?

I am unable to derive it due to the expresion involving (a, b) on Right hand side.

Answer 1

Pls recall how the standard inner product in $\mathbb C^n$ is defined: $$\big((x_1,\dots,x_n)\mid (y_1,\dots,y_n)\big) \:=\: \sum_{i=1}^nx_i\,\overline{y_i}$$ It involves complex conjugation of one of the two arguments, and which one is chosen is a matter of convention. Hence your obstacle is not a lack of derivation, but depends on the definition only.

There is a tendency to have the second argument slot conjugate-linear in maths, whereas in physics one often encounters conjugate linearity in the first argument (cf the bra-ket-notation in this context).