Page 358 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

In Section 1.3 we introduced the concept of a linear combination of column vectors. The following definition extends this
idea to more general vectors.

DEFINITION

A vector w is called a linear combination of the vectors if it can be expressed in the form

where are scalars.

Remark If , then the equation in the preceding definition reduces to ; that is, w is a linear combination of a
single vector if it is a scalar multiple of .

EXAMPLE 8 Vectors in Are Linear Combinations of i, j, and k

Every vector in is expressible as a linear combination of the standard basis vectors

since

EXAMPLE 9 Checking a Linear Combination

Consider the vectors and in . Show that is a linear combination of u and v and

that is not a linear combination of u and v.

Solution ; that is,

In order for w to be a linear combination of u and v, there must be scalars and such that
or
Equating corresponding components gives

Solving this system using Gaussian elimination yields , , so