Understanding $\mathbb{K}$ and $\mathbb{R}^n$ - Get Your Answers Here

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In summary, the symbols $\mathbb{K}$ and $\mathbb{R}^n$ are often used to represent an arbitrary field and n-dimensional real space, respectively. The former may also refer to a vector space or other structures depending on the context. The latter is defined as the set of n-tuples of real numbers, but can be considered as a vector space with a specific convention for operations. Additional structures such as algebraic or topological properties may also be assumed.
  • #1
Hernaner28
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Hi. What does that mean when we see:

$$\mathbb{K}$$ what set is that? definitely not the reals, integers, etc.
and
$$\mathbb{R}^n$$ Is that the reals or what?

Thanks!
 
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  • #2
the blackboard K is often used to refer to an arbitrary field. If you encounter it somewhere it should be stated what is meant by it.

The blackboard R^n just means n-dimensional real space. So basically (up to linear isomorphism) the unique n-dimensional vector space over the real numbers.
 
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Thank you!
 
  • #4
conquest said:
the blackboard K is often used to refer to an arbitrary field. If you encounter it somewhere it should be stated what is meant by it.

The blackboard R^n just means n-dimensional real space. So basically (up to linear isomorphism) the unique n-dimensional vector space over the real numbers.
A tiny disagreement. R^n is the set of order n-tuples of real numbers. It becomes a vector space only with the convention that "sum" and "scalar multiplication" are "coordinate wise". Yes, that is the "natural" convention but it separate from just "R^n".
 
  • #5
You are right of course. Naturally my response was in terms of the vector space since this is the first definition I saw. You could also assume this and express that you are only talking strictly as the n-tuples of real numbers by mentioning you use the bare set underlying the vector space. I think in most literature the vector space (indeed also the natural topological, norm, inner product, Lie group and manifold (symplectic, smooth, Riemannian)) structure are assumed when used. A thing that might be stated explicitely is any algebra structure.

So immediately when I see that symbol these things I also assume, but it might pay to be a bit more reserved about this.
 

FAQ: Understanding $\mathbb{K}$ and $\mathbb{R}^n$ - Get Your Answers Here

What is the difference between $\mathbb{K}$ and $\mathbb{R}^n$?

The set $\mathbb{K}$ is the set of all complex numbers, while $\mathbb{R}^n$ is the set of all n-dimensional real vectors. In other words, $\mathbb{K}$ contains both real and imaginary numbers, while $\mathbb{R}^n$ only contains real numbers.

What are some common applications of understanding $\mathbb{K}$ and $\mathbb{R}^n$?

Understanding $\mathbb{K}$ and $\mathbb{R}^n$ is important in various fields such as physics, engineering, and computer science. It is used in calculations involving complex numbers, vectors, and matrices.

How do I visualize $\mathbb{K}$ and $\mathbb{R}^n$?

One way to visualize $\mathbb{K}$ is using the complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. For $\mathbb{R}^n$, you can imagine a Cartesian coordinate system with n axes, where each axis represents a different dimension.

What is the significance of the dimension in $\mathbb{R}^n$?

The dimension in $\mathbb{R}^n$ represents the number of coordinates needed to describe a point in n-dimensional space. For example, in $\mathbb{R}^2$, a point is described using two coordinates (x and y), while in $\mathbb{R}^3$, a point is described using three coordinates (x, y, and z).

How does understanding $\mathbb{K}$ and $\mathbb{R}^n$ relate to linear algebra?

Linear algebra is the study of vector spaces, and both $\mathbb{K}$ and $\mathbb{R}^n$ are examples of vector spaces. Understanding these sets is crucial in solving systems of linear equations, finding eigenvalues and eigenvectors, and performing matrix operations.

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