We define a linear combination of vectors and examine whether a given vector may be expressed as a linear combination of other vectors, both algebraically and geometrically.

VEC-0040: Linear Combinations of Vectors

For example, is a linear combination of , and because

In this module we will focus on vectors in and .

Geometry of Linear Combinations

The method we used in Example ex:lincombparallelogrammethod to express the given vector as a linear combination of two other vectors is sufficiently useful that we summarize the steps.

Algebra of Linear Combinations

Practice Problems

Solve a system of linear equations to express as a linear combination of and .

System of linear equations:

Values of and :

Linear Combination:

Use Procedure pro:lincombgeo to express as a linear combination of and .

Linear Combination:

Use two different approaches (algebraic and geometric) to explain why the vector cannot be expressed as a linear combination of vectors and .
Set up a system of equations that can be used to express as a linear combination of vectors , and .

System of Equations: