Basis Vectors

Introduction

In the realm of linear algebra, basis vectors play a crucial role in defining vector spaces. A basis is a set of vectors that are linearly independent and span the vector space. Basis vectors provide a foundation for representing any vector in the vector space as a linear combination of these basis vectors.

Definitions

1. Basis: A basis for a vector space is a set of vectors that are linearly independent and span the vector space. Given a vector space $V$, a set of vectors $B=\{v_1,v_2,...,v_n\}$ is a basis for $V$ if every vector $v\in V$ can uniquely be expressed as a linear combination of the basis vectors $v_1,v_2,...,v_n$.

2. Basis Vectors: The vectors that form a basis for a vector space. These vectors are chosen such that they are linearly independent and span the entire vector space.

Explanation

Basis vectors act as building blocks for creating and navigating within a vector space. Each vector in the space can be uniquely identified by a specific combination of the basis vectors. This property simplifies operations within the vector space, making it easier to perform calculations and understand the structure of the space.

In ${\mathbb{R}}^2$, the standard basis vectors are ${\vec{e}}_1=\left[\begin{array}{c}1\\ 0\end{array}\right]$ and ${\vec{e}}_2=\left[\begin{array}{c}0\\ 1\end{array}\right]$. Any vector in ${\mathbb{R}}^2$ can be expressed as $a{\vec{e}}_1+b{\vec{e}}_2$, where $a$ and $b$ are scalars.

In ${\mathbb{R}}^3$, the standard basis vectors are ${\vec{e}}_1=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$, ${\vec{e}}_2=\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]$, and ${\vec{e}}_3=\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$. Any vector in ${\mathbb{R}}^3$ can be expressed as $a{\vec{e}}_1+b{\vec{e}}_2+c{\vec{e}}_3$, where $a$, $b$, and $c$ are scalars.

Historical Context

The concept of basis vectors dates back to the early development of linear algebra in the 19th century. Mathematicians like Georg Cantor, David Hilbert, and Hermann Grassmann contributed to the foundational principles of vector spaces and basis vectors. The idea of basis vectors revolutionized the understanding of vector spaces and paved the way for advanced mathematical applications in various fields.

Real-life Examples

• In computer graphics, basis vectors are used to represent the orientation of 3D objects and manipulate their positions in space.
• In quantum mechanics, basis vectors are essential for describing the states of quantum systems and performing calculations related to quantum properties.

Exam Questions

1. Define basis vectors and explain their significance in vector spaces.
2. Find the basis vectors for the vector space ${\mathbb{R}}^4$.
3. How are basis vectors used in linear transformations?

Remember, understanding basis vectors is fundamental in the study of linear algebra and plays a vital role in various mathematical applications.