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Every Vector Space has a Basis

By Moloy De posted Sat October 22, 2022 02:40 AM

  
The set of real numbers is one dimensional and real numbers are scalers. When one considers the tuples of real numbers the underlying set is multidimensional and the tuples are called vectors. A vector has a length which is the norm of the vector and the directions.

Consider two vectors. The linear combinations of them is again a vector space that can be proved mathematically starting from a definition of a vector space. Converse is true also. Given a vector space one can find vectors whose linear combinations span the vector space. The method is constructive. One can start with an arbitrary vector in the vector space and go on choosing vectors until the collection spans the entire vector space. Since choosing dependent vector doesn't include any new vector in the span, one choses vectors that are linearly independent or orthogonal.

The maximum number of linearly independent vectors is the dimension of the vector space and the collection of those linearly independent vectors is called the basis of the vector space. The existence of a basis of an arbitrary vector space needs a mathematical proof that uses Zorn's Lemma. The proof is available here.

QUESTION I: Provide an example of an infinite dimensional vector space.
QUESTION II: How does rank of a matrix is related to the dimension of a vector space?
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Ken Iida
Fri April 07, 2023 03:12 AM

Answer 1 
For example, a linear space created by functions that appears in functional analysis.