金融工程课件3约翰郝尔john hull.pdf

Financial Engineering and Risk Management Review of vectors Martin Haugh Garud Iyengar Columbia University Industrial Engineering and Operations Research Reals numbers and vectors We will denote the set of real numbers by Vectors are finite collections of real numbers Vectors come in two varieties Row vectors: v = v1 v2 . . . vn   w1 w   2   Column vectors w = .  .  . wn By default, vectors are column vectors The set of all vectors with n components is denoted by n 2 Linear independence A vector w is linearly dependent on v1 , v2 if w = α v + α v for some α , α ∈ 1 1 2 2 1 2 Example:       2 1 0       6 = 2 1 + 4 1 4 0 1 Other names: linear combination, linear span A set V = {v1 , . . . , vm } are linearly independent if no vi is linearly dependent on the others, {vj : j = i} 3 Basis Every w ∈ n is a linear combination of the linearly independent set    