Vector Space Basis

A Vector Space Basis is a vector set in a vector space V such that the vectors are linearly independent and every vector in the vector space is a linear combination of this set.

• AKA: Vector Basis, Basis.
• Example(s):
  • Orthonormal Basis,
  • Hamel Basis,
  • Hilbert Basis,
  • Schauder Basis.
  • …
• Counter-Example(s):
  • Basis Vector,
  • Vector Norm.
• See: Dimension (Vector Space), Tensor, Vector Space, Linearly Independent, Linear Combination, Spanning Set, Vector Span, Linear Independence.

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References

2020a

• (Wikipedia, 2020) ⇒ https://en.wikipedia.org/wiki/Basis_(linear_algebra) Retrieved:2020-8-15.
  • In mathematics, a set of elements (vectors) in a vector space V is called a basis, if every element of V may be written in a unique way as a (finite) linear combination of elements of . The coefficients of this linear combination are referred to as components or coordinates on of the vector. The elements of a basis are called basis vectors.

    Equivalently is a basis if its elements are linearly independent and every element of is a linear combination of elements of . In more general terms, a basis is a linearly independent spanning set.

    A vector space can have several bases; however all the bases have the same number of elements, called the dimension of the vector space.

2020b

• (Todd, 2020) ⇒ Rowland, Todd. “Basis." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Basis.html
  • QUOTE: The word basis can arise in several different contexts. Speaking in general terms, an object is "generated" by a basis in whatever manner is appropriate.

    For example, a vector space can have a vector basis which spans the vector space by finite linear combinations.

2020c

• (Todd, 2020) ⇒ Rowland, Todd and Weisstein, Eric W. “Vector Basis." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/VectorBasis.html
  • QUOTE: A vector basis of a vector space $V$ is defined as a subset $v_1,\cdots,v_n$ of vectors in $V$ that are linearly independent and span $V$. Consequently, if $(v_1,v_2,\cdots,v_n)$ is a list of vectors in $V$, then these vectors form a vector basis if and only if every $v$ in $V$ can be uniquely written as
    $v=a_1v_1+a_2v_2+\cdots+a_nv_n$, (1)

    where $a_1,\cdots, a_n$ are elements of the base field.

    When the base field is the reals so that $a_i$ in $R$ for $i=1,\cdots,n$, the resulting basis vectors are n-tuples of reals that span n-dimensional Euclidean space $R^n$. Other possible base fields include the complexes $C$, as well as various fields of positive characteristic considered in algebra, number theory, and algebraic geometry.

    A vector space $V$ has many different vector bases, but there are always the same number of basis vectors in each of them. The number of basis vectors in $V$ is called the dimension of $V$. Every spanning list in a vector space can be reduced to a basis of the vector space.

    The simplest example of a vector basis is the standard basis in Euclidean space $R^n$, in which the basis vectors lie along each coordinate axis. A change of basis can be used to transform vectors (and operators) in a given basis to another.

2015

• (Lake Tahoe Community College, 2015) ⇒ http://ltcconline.net/greenl/courses/203/Vectors/basisDimension.htm
  • Basis: In our previous discussion, we introduced the concepts of span and linear independence. In a way a set of vectors S = {v1, ..., vk} span a vector space V if there are enough of the right vectors in S, while they are linearly independent if there are no redundancies. We now combine the two concepts.
  • Definition of Basis: Let V be a vector space and S = {v1, v2, ..., vk} be a subset of V. Then S is a basis for V if the following two statements are true.
    • 1. S spans V.
    • 2. S is a linearly independent set of vectors in V.
  • We have seen that any vector space that contains at least two vectors contains infinitely many. It is uninteresting to ask how many vectors there are in a vector space. However there is still a way to measure the size of a vector space. For example, R3 should be larger than R2. We call this size the dimension of the vector space and define it as the number of vectors that are needed to form a basis. Tow show that the dimensions is well defined, we need the following theorem.

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