What is the Cardinality and Dimension of \mathbb{Z}^{3}_{7}?

In summary, cardinality and dimension are two different measures used to describe sets and spaces. Cardinality represents the number of elements in a set, while dimension represents the number of independent variables needed to describe a space. In set theory, cardinality is denoted by the symbol "ℵ" and is often referred to as the order or size of a set. The maximum cardinality of a set is known as its "power set" and is equal to 2 to the power of the cardinality of the original set. Dimensionality, on the other hand, is used in data analysis to refer to the number of features or variables used to describe a dataset. It is important to consider dimensionality when analyzing data to avoid overfitting
  • #1
jdstokes
523
1
Find the cardinality and dimension of the vector space [itex]\mathbb{Z}^{3}_{7}[/itex] over [itex]\mathbb{Z}_{7}[/itex].

[itex]\mathbb{Z}^{3}_{7} = \{ (a,b,c) \; | \; a,b,c \in \mathbb{Z}_{7} \}[/itex].

Then since [itex]\mathbb{Z}_{7}[/itex] is a field [itex]1 \cdot a = a \; \forall \; a[/itex], so [itex]B = \{ (1,0,0), (0,1,0) , (0,0,1) \}[/itex] is a basis of [itex]\mathbb{Z}^{3}_{7}[/itex], so [itex]\dim \mathbb{Z}^{3}_{7} = 3[/itex]. ans = 9, what the?

Thanks

James
 
Last edited:
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  • #2
Hi again,

Does anyone have any clues on this one? I'm really stuck.

Thanks

James.
 
  • #3
Are you sure you copied the problem correctly? Because 3 certainly seems to be the right answer.
 

FAQ: What is the Cardinality and Dimension of \mathbb{Z}^{3}_{7}?

1. What is the difference between cardinality and dimension?

Cardinality refers to the number of elements in a set, while dimension refers to the number of independent variables needed to describe a space. In other words, cardinality is a measure of size, while dimension is a measure of the complexity of a space.

2. How is cardinality represented in set theory?

In set theory, cardinality is represented by the cardinal numbers, which are denoted by the symbol "ℵ". The cardinality of a set is often referred to as its "order" or "size".

3. What is the maximum cardinality of a set?

The maximum cardinality of a set is known as its "power set" and is equal to 2 to the power of the cardinality of the original set. For example, if a set has a cardinality of 3, its power set would have a cardinality of 2^3 = 8.

4. How is dimensionality used in data analysis?

In data analysis, dimensionality refers to the number of features or variables used to describe a dataset. It is important to consider dimensionality when analyzing data, as having too many variables can lead to overfitting and decrease the accuracy of the analysis.

5. Can the dimension of a space be greater than its cardinality?

Yes, it is possible for the dimension of a space to be greater than its cardinality. This can occur when there are dependent variables or when the space is infinite. For example, a line in 3-dimensional space would have a dimension of 1, but a cardinality of infinity.

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