The term linear combination is fundamental to linear algebra and will be used throughout this text. A linear combination of a set of vectors can be defined as the addition of these vectors scaled by a corresponding ordered set of scalar coefficients : For example, let's consider the following 3 vectors: In this case, is a linear combination of and . Close inspection shows that : This linear combination is illustrated graphically in Figure 2-4 where you can see that is composed of 1 and 3 's.
Object could not be loaded. Figure 2-4.
Geometry of a Linear Combination
Linear combinations will often be used to define more complex mathematical sets or geometric objects. For example, a line in is defined as the combination of a starting vector (in this case ) with a direction vector ( ) which is scaled by a "free parameter" . The term free parameter simply states that the scalar value is free to take on any real value between positive and negative infinity or in interval notation . Figure 2-5 illustrates how this linear combination maps out a line in .
Object could not be loaded.
Figure 2-5.
Parameterized Line ( )