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".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.
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.
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.
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.
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).
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.