Abstract
In this chapter, we learn the notions of “subspace generation” and “linear independence”, which play essential roles in linear algebra. These yield the notions of “basis” and “dimension”, and they turn out to be useful tools to analyze a linear space. Some important theorems concerning them will appear. The notion of linear independence is very important, and the readers are encouraged to understand its meaning to proceed to the next step.
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Notes
- 1.
The existence of such a subspace was shown at the end of Sect. 2.2.
- 2.
We showed that \(x\boldsymbol{a}=\boldsymbol{0}\Rightarrow x=0\) for \(\boldsymbol{a}\not =\boldsymbol{0}\) in Sect. 2.1.
- 3.
The linear independence and the linear dependence do not depend on the order of arrangement of vectors, so by rearranging the vectors we may assume \(x_1\not =0\).
- 4.
“Randomly” means here that we take two vectors independently according to the three-dimensional standard normal distribution (this “independence” is in the sense of probability theory and is not “linear independence”). We may think that two points are chosen arbitrarily from the points in Fig. 3.4 (right) in Sect. 3.6.
- 5.
First choose \(\vec {a}\) randomly, then it is not the origin with probability 1. Next choose \(\vec {b}\) randomly, then it is not on the line passing through \(\vec {a}\) and the origin with probability 1.
- 6.
If we do not call the function seed, the seed is taken from the internal clock of our computer. We get different results each time we run it.
- 7.
Depending on the seed, one vector may look close to the zero vector or two vectors may look overlapped.
- 8.
As stated in Sect. 3.1, \(\{\}\) is the smallest set generating \(\{\boldsymbol{0}\}\).
- 9.
As we defined in Sect. 3.1, V is finite dimensional if it is generated by a finite number of vectors.
- 10.
See the supplementary remark on infinite dimension in Sect. 3.6.
- 11.
In this module functions useful for various calculations on matrices are defined. We will learn them in Chap. 5.
- 12.
Like the built-in function print, the function f can have any numbers of arguments. For example, if we call it in the format f(a, b, c) for three arguments, the function matrix_rank is called in the format matrix_rank([a, b, c]) for one argument of the list consisting of a, b, and c. If we omit * before the formal argument x in the definition of f, we need to call it in the format f([a, b, c]).
- 13.
We sometimes distinguish the direct sum of subspaces as the internal direct sum and the direct sum of spaces as the external direct sum. However, they are essentially the same.
- 14.
The image drawn by VPython is actually projected on the computer screen which is a two-dimensional plane. We feel that it is drawn in three-dimensional space as we change the viewpoint with the mouse. It is one of the important themes after Chap. 6 of this book to project a space with large dimension, not only 3D, into a smaller space (plane, straight line).
- 15.
For objects of a class whose elements can be specified by subscript such as lists, tuples, and strings, slice is a mechanism referring to their elements.
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Tsukada, M., Kobayashi, Y., Kaneko, H., Takahasi, SE., Shirayanagi, K., Noguchi, M. (2023). Basis and Dimension. In: Linear Algebra with Python. Springer Undergraduate Texts in Mathematics and Technology. Springer, Singapore. https://doi.org/10.1007/978-981-99-2951-1_3
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DOI: https://doi.org/10.1007/978-981-99-2951-1_3
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