## Multiple Regression

## 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: ## Linear Combinations

32351351

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
pointin a P-dimensional space. The elements of the list are the coordinates of the point. The observationAis represented by a point.- A
directed line segment(avector, anarrowor aray) in a P-dimensional space. ObservationsB, CandDare represented by arrows.- A
linear combinationof 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.