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This question comes from this post.

This figure is trying to illustrate 4 spaces defined by 4 different set of standard basis.

In mathematics, the standard basis (also called natural basis) for a Euclidean space is the set of unit vectors pointing in the direction of the axes of a Cartesian coordinate system.

Is "standard basis" here countable or not? I guess not and need a double check.

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The very quote in the question answers the question:

...4 spaces defined by 4 different set of standard basis.

or more correctly in my view:

...4 spaces defined by 4 different sets of standard bases.

"Basis" is countable, the plural is "bases" as shown by this disctionary.comn link winch gives as sense 5:

Mathematics. a set of linearly independent elements of a given vector space having the property that every element of the space can be written as a linear combination of the elements of the set.

This Macmillan entry also shows it as countable, as does this merriam-webster definition

Now while it is true that for any one space there is only one standard basis, there are different bases for different spaces. In any case "basis" is countable, even if in a particular use only one standard basis exists.

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There are a couple of issues in the phrasing:

1. A standard basis is already a set

from Wikipedia:

The standard basis (also called natural basis) for a Euclidean space is the set of unit vectors pointing in the direction of the axes of a Cartesian coordinate system.

Therefore, "sets of standard basis" would imply multiple groupings of different standard basis. However, the OP is referring to a pair of vectors as a "set of standard basis", which is incorrect because the standard basis is already that set of vectors. Therefore a more appropriate phrasing would be:

4 different standard bases

Where "bases" is the plural of basis, pronounced like \ˈbā-​ˌsēz\ and not \ˈbā-​səz\


That is not to say you cannot have "sets of bases" but that is not what is in the OP's post.

2. There is only one Standard Basis in any space

The issue is, only the upper left graph is showing the standard basis for the real plan. The rest are just sets that make a orthogonal unit basis of the real plane.

A standard basis is the specifically ordered set of unit vectors ei, with 1 in the ith coordinate:

R2 (1,0); (0,1)

R3 (1,0,0); (0,1,0); (0,0,1)

R4 (1,0,0,0); (0,1,0,0); (0,0,1,0); (0,0,0,1)

etc.

standard basis is not appropriate ot refer to those four diagrams because only one of them is the standard basis for R2. See wikipedia example.


Therefore, the grammatically and mathematically correct way to phrase this statement could be:

This figure is trying to illustrate 4 spaces defined by 4 different bases.

It could be made more precise by adding the adjective "orthogonal" and "unit" would be appropriate. (Also note that the OP goes on to say that all 4 spaces are the same space, which is correct)

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