Notes on Differential Geometry , 1 Lars Andersson

1.1. Linear Algebra. Let F = R or F = C and let V be a vector space over F. A finite dimensional vector space over R (C) is isomorphic to Rn (Cn) for some n, where Rn = {x = (x1, . . . , xn)} for xi ∈ R and Cn = {z = (z1, . . . , zn)} for zi ∈ Cn. On Rn, the standard Euclidean inner product is given by 〈x, y〉 = ∑ xiyi and the Euclidean norm is given by ||x|| = 〈x, x〉 2 . On Cn the standard Hermitean structure is given by 〈z, w〉 = ∑ ziw̄i and similarly the corresponding norm is given by ||z|| = 〈z, z〉 1 2 . Note that in the case of the Hermitean structure the inner product is a sesquilinear form, i.e. 〈az, w〉 = a〈z, w〉 and 〈z, aw〉 = ā〈z, w〉 for a ∈ C. Let V,W be vector spaces. A mapping A : V → W is said to be linear if for all a, b ∈ F and v, w ∈ V ,