§1
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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