Vector Space Basis

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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.



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

2020c

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.