Vectors

A vector is a collection of numbers arranged in a column or a row and can be thought of as a point in space. We review basic notions such as independence, span, subspaces, and dimension. The scalar product allows to define the length of a vector, as well as generalize the notion of the angle between two vectors. Via the scalar product, we can view a vector as a linear function. We can also compute the projection of a vector onto a line defined by another — a basic ingredient in many visualization techniques for high-dimensional data sets.

Outline

• Basics
• Scalar product, norms and angles
• Projection on a line
• Orthogonalization
• Hyperplanes and half-spaces
• Linear functions
• Application: data visualization via projection on a line
• Exercises