Linear Combination and Span I

Introduction

Summary   As a means to create new vectors from a set of known ones we introduce the concepts of linear combination and span. We develop some basic properties and will revisit this topic from a higher perspective later.

Outline   Often we will find ourselves in a situation where we have one or more distinguished vectors in \(\RNrSpc{n}\) and need to use them to construct new vectors. This is possible by forming ‘linear combinations’. Let's look a this in detail: say the distinguished vectors are \(\Vect{s}_1,\dots ,\Vect{s}_r\). So, we have \(r\) such. A linear combination of these results from this two-step procedure:

1. First we rescale each of these vectors by any numbers of our choosing. So, \(\Vect{s}_1\) gets rescaled by some number \(t_1\) to yield \(t_1\Vect{s}_1\). Then \(\Vect{s}_2\) gets rescaled by some number \(t_2\) to yield \(t_2\Vect{s}_2\), and we keep rescaling these vectors until, finally \(\Vect{s}_r\) gets rescaled by some number \(t_r\) to yield \(t_r\Vect{s}_r\).

2. Then we sum the rescaled vectors to obtain

   \[\Vect{x} = t_1\Vect{s}_1+ \cdots + t_r\Vect{s}_r\]

   Now, we have one linear combination of the vectors \(\Vect{s}_1,\dots ,\Vect{s}_r\).

Every time we choose new rescaling factors \(t_1,\dots ,t_r\) for the vectors \(\Vect{s}_1,\dots ,\Vect{s}_r\), we obtain a new linear combination of the distinguished vectors.

Span   The collection of all possible linear combinations is the span of the given vectors. As a first application, span allows us to describe parallelograms and parallelepipeds in \(\RNrSpc{n}\).

Colors via linear combinations   As a practical application of forming linear combinations and spans of vectors, here is how they help us define colors on a computer screen.