Linear Algebra - Inner Product
maths ✒ linear algebra ✒ inner product
Given a vector space, V, an inner product on V is a function $ V^2 \rightarrow \mathbb{C} $ satisfying certain conditions. The inner product of u, v is written $ \langle u, v \rangle $
Conditions:
- linearity in the 1st variable: $ \langle \mu x + \lambda y, z \rangle = \mu \langle x, z \rangle + \lambda \langle y, z \rangle $
- conjugate symmetry: $ \langle u, v \rangle = \overline{\langle v, u \rangle}
- $<0,0> = 0 $ and $ < v,v > \gt 0 \forall v \ne 0 $ (positive definite)
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