Right now, I am reading a text on Nonlinear Optimization for a class, and there is a minimization example where for $A$, a matrix of dimension $m \times n$, $c \in \mathbb{R}^{n}$ and $b \in \mathbb{R}^{m}$, we have that the Lagrangian is given by $$L(x, \lambda) = \langle c, x \rangle + \langle \lambda, b-A x \rangle = \langle c - A^{T}\lambda, x \rangle + \langle b, \lambda \rangle $$
where $A^{T}$ denotes the transpose of the matrix $A$.
I know that there are inner product rules for transposes that allow us to have $ \langle c, x \rangle + \langle \lambda, b-A x \rangle = \langle c - A^{T}\lambda, x \rangle + \langle b, \lambda \rangle$, but I do not know what they are.
Could someone please explain to me why this equality is true, as well as listing the properties of inner products that I would need in order to tackle problems like this in the future (or at least some place online where I can find all of these properties)?
Thank you.