Bázisvektorok

Bázisvektorok, in English known as basis vectors, are fundamental concepts in linear algebra and vector spaces. They form a set of linearly independent vectors that can be used to represent any other vector within that space as a unique linear combination. Think of them as the building blocks from which all other vectors in a given space can be constructed.

The most common example of basis vectors is found in a two-dimensional Cartesian coordinate system. Here, the standard basis vectors are usually denoted as i and j. The vector i points along the positive x-axis and has a magnitude of 1, represented as (1, 0). The vector j points along the positive y-axis and also has a magnitude of 1, represented as (0, 1). Any vector in this 2D plane, such as (a, b), can be uniquely expressed as a linear combination of i and j: a*i + b*j.

In three dimensions, the standard basis vectors are i, j, and k, corresponding to the x, y, and z axes respectively. These are (1, 0, 0), (0, 1, 0), and (0, 0, 1). Any vector (a, b, c) in 3D space can be written as a*i + b*j + c*k.

A crucial property of basis vectors is that they are linearly independent, meaning no vector in the set can be expressed as a scalar multiple of another. The number of vectors in a basis is called the dimension of the vector space. For example, the standard basis vectors i and j form a basis for a 2-dimensional space.