Abstract
Although the content in this chapter should be taught in much earlier life stages, graduate or undergraduate levels, of most possible reader, because this book mainly deals with somewhat data science oriented matters, it might not be a bad idea to reintroduce fundamental concepts in a data science oriented manner.
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Notes
- 1.
Practically, employing only the first scalars in RGB representation is equivalent to the usage of red sunglass through which only red color can penetrate. Now, colors are transformed to real values that describe red color intensity of colors, although in this example only integers are allowed since colors are treated as an example dummy scalars.
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Appendix
Appendix
1.1 I Rank
In this appendix, rank of matrix is briefly introduced because the concept of rank is important in the next chapter. Suppose that matrix \(X \in \mathbb {R}^{N \times M}\) is represented as M N-dimensional vectors, \(\boldsymbol {x}_j \in \mathbb {R}^N\), as
If there are vectors, \(\boldsymbol {c}_j \in \mathbb {R}^{M'}, 1 \leq j \leq M\), such that
where J is a set of M′ integers taken from [1, M] without repetitions; the smallest M′ is called as the rank of matrix, otherwise the rank of matrix is equal to M. In other words, not all x js are independent but at most M′ out of M x js are independent. This means that x js span not M dimensional space but at most M′(< M) dimensional space. Thus, the rank of tensor is at most min(M,N) because the number of independent vectors cannot exceed the number of dimensions.
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Taguchi, Yh. (2020). Introduction to Linear Algebra. In: Unsupervised Feature Extraction Applied to Bioinformatics. Unsupervised and Semi-Supervised Learning. Springer, Cham. https://doi.org/10.1007/978-3-030-22456-1_1
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