Forrest Young's Notes
Copyright © 1997-9 by Forrest W. Young.
Linear Combinations: Example
Much of multivariate analysis is concerned with linear combinations of variables. Consider the following three observations on two variables:
X1 X2 obs1 2 4 obs2 5 2 obs3 1 2 We can create a new variable Y that is a linear combination of the two X's by multiplying each of the original variables by a constant and adding the products.
Let's say we arbitrarily select mutiplicands of 3 and 2. Then we have:
Y = 3(X1) + 2(X2)
And we have
Y = [14 19 7]
Here is a schematic of the geometry of the linear combination above. Notice that the result of the linear combination Y, is shown, both in the schematic and in the table below it.
Geometry of Linear Combinations
X1 X2 Y obs1 2 4 14 obs2 5 2 19 obs3 1 2 7