Bra-ket notation and other linear algebra stuff

[iScience]

forgive the messiness; i take bad notes in class.

http://i.imgur.com/VmW8Ubg.jpg

towards the middle of the page where it says "this is equivalent to..." and then my professor wrote what follows but i thought the row vector should be complex conjugates? ie, the red writings are not actually part of the notes, i just wrote them because i was confused as to why they weren't there.
http://i.imgur.com/5zGGBe3.jpg

towards the bottom right, where it has a(hermitian)IbI, which are all matrices by the way, i don't understand where the second 'I' (unit matrix) came from (where i stated my confusion in the notes).btw, does anyone know how to shrink an image when i do ['img']link['/img']? sometimes these images are too large

 

[JorisL]

Your first remark is correct. They should be complex conjugates.

For the second part if you have numbers like $$a\cdot u^{-1} \cdot u \cdot b$$.
Than you get $$a\cdot 1\cdot b$$. you can freely multiply this entire expression with 1.

I believe the second unit matrix is added to convert the vector b in the transposed vector b since otherwise you have the product of 2 column vectors.

 

[iScience]

towards the bottom of the page where i wrote "where'd this one come from?", to the RHS of that, i have 'ab'.

but if this is to become < a | b > , then shouldn't the 'a' in 'ab' be hermitian?