Forrest Young's Notes
Copyright © 1997-9 by Forrest W. Young.
A central aspect of Multiple regression is the idea of a linear combination. In this section we illustrate the idea of linear combinations with the following "dataset" of four observations on two variables:
3 2 3 5 1 3 5 1
- Algebraically, a list of numbers can represent:
- A numeric variable in a dataset (a column) -- This is often called a column vector. Thus, [3 3, 1, 5] is a column vector in these data.
- An observation in a dataset (a row) -- often called a row vector. Thus, [3, 5] is a row vector in these data.
- Geometrically, we can portray Linear combinations as shown in the following diagram:
Diagram: In this diagram, a list of P numbers can represent:
- A point in a P-dimensional space. The elements of the list are the coordinates of the point. The observation A is represented by a point.
- A directed line segment (a vector, an arrow or a ray) in a P-dimensional space. Observations B, C and D are represented by arrows.
- A linear combination of P variables. The elements of the list are the coefficients of the linear combination. This is not represented in this diagram. We'll get to it in a minute.
Note: There are two different meanings of the word "vector" here.
- Algebraically, a list of numbers.
- Geometrically, a directed line segment.