How would I go about expanding the inner product $$\langle av+bw,cx+dy\rangle?$$ I'm confused since you can only add to one side at a time. For example it makes sense to me that $$\langle av+bw,x\rangle=a\langle v,x\rangle+b\langle w,x\rangle,$$ but I'm confused as to how it's possible to have multiple terms as part of both terms.
Thanks!
Since you understand the idea of linearity in the first argument, all you have to do is compute the bilinearity sequentially, as follows $$ \langle av+bw,cx+dy\rangle=a\langle v,cx+dy\rangle + b\langle w,cx+dy\rangle $$ Then now, consider linearity with the second argument, which gives $$ \langle av+bw,cx+dy\rangle=ac\langle v,x\rangle + ad\langle v,y\rangle+bc\langle w,x\rangle + bd\langle w,y\rangle $$