In the previous segments, you have learnt how to represent vectors and matrices and understood some of their important operations. We will dive into one of the most fundamental building blocks of PCA: Basis. But before we get into the math part of it, let’s understand, in a very intuitive way, what it represents, in the following lecture.
Essentially, ‘basis’ is a unit in which we express the vectors of a matrix.
For example, we describe the weight of an object in terms of kilogram, gram and so on; to describe length, we use a metre, centimetre, etc. So for example, when you say that an object has a length of 23 cm, what you are essentially saying is that the object’s length is 23×1 cm. Here, 1 cm is the unit in which you are expressing the length of the object.
Similarly, vectors in any dimensional space or matrix can be represented as a linear combination of basis vectors.
Let’s discuss them in further detail in the following lecture.
Let’s unpack the ideas that you learnt in the above video. Since i and j themselves represent (1,0) and (0,1), you can represent any vector in the 2-D space with these i and j vectors.
Any vector ‘a’ (ax,ay) can be represented in a 2-D space, using the following notation:
a=axi+ayj
or
a=ax⋅[10]+ay⋅[01]
Visually, it can be represented as follows:
For example, a vector A (2,3) can be written as 2⋅[10]+3⋅[01]. In order to obtain the vector A, we scaled i by 2 and j by 3 and then finally added them up.
This scaling and adding the vectors up to obtain a new vector is also known as a linear combination.
For the patients’ dataset that we had earlier, we can denote each patient vector by the following notation:
Therefore, now we can say that Patient 1 is represented by 165(1 cm,0) + 55(0,1kg). And similarly, we can express other patients’ information as well.
The basic definition of basis vectors is that they’re a certain set of vectors whose linear combination is able to explain any other vector in that space.
In a 2D space, the standard basis vectors are given by [10] and [01]. In a 3D space, the same are given by ???100???, ???010??? and ???001???. As you can see, an n-dimensional space or a dataset having n variables would have n standard basis vectors.
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