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Chapter 3 Linear algebra with applications in biology

§3.1 Basic operations of vectors

A vector [pic] space is a triple [pic] where [pic] is the addition and [pic] is a scalar multiplication. First we introduce the [pic]-dimensional Euclidean space [pic], one of the most important vector spaces.

Let

[pic].

Define the addition of two vectors [pic] and [pic] as following:

[pic], [pic]

[pic].

The element [pic] is the zero element of [pic]. Obviously [pic] for all [pic]. The geometric meaning of addition of two vectors is described as the following figure for case [pic]:

[pic]

Figure 3.1.1

We define the scalar multiplication in [pic] as following. Let [pic], [pic]. Define

[pic].

The geometric meaning of [pic] is described as following

[pic]

Figure 3.1.2

Next we introduce the inner product of two vectors [pic], [pic]. Define

[pic] or [pic]

The length of the vector [pic] is defined as

[pic]

which is the distance between [pic] and the origin [pic]. From law of cosine, we have (see Fig 3.1.3)

[pic].

On the other hand, we have

[pic][pic]

[pic]

Hence

[pic]

Obviously if [pic], i.e., [pic] then [pic]

[pic]

Fig 3.1.3

Example 1.1: Find the projection of vector [pic] on the vector [pic].

Ans: [pic]

[pic]

[pic]

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[pic]

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