- #1
pivoxa15
- 2,255
- 1
Homework Statement
What are the differences between the two?
quasar987 said:Just surfing wiki, I get the impression that there is none. Do you have a reason to believe that there is a difference?
A free basis, also known as a basis of a vector space, is a set of linearly independent vectors that span the entire vector space. In other words, any vector in the space can be expressed as a linear combination of the vectors in the basis.
The terms "free basis" and "basis" are often used interchangeably, but there is a subtle difference. A free basis refers specifically to a basis of a vector space, while a basis can also refer to a basis of a module or other algebraic structure.
To determine if a set of vectors is a free basis, you can use the following two criteria: 1) the vectors must be linearly independent, meaning that no vector in the set can be expressed as a linear combination of the other vectors, and 2) the vectors must span the entire vector space, meaning that every vector in the space can be expressed as a linear combination of the vectors in the set.
Yes, a vector space can have multiple free bases. This is because a vector space can have infinitely many linearly independent sets of vectors that span the space. However, all free bases of a given vector space will have the same number of vectors, known as the dimension of the space.
Free bases are an important concept in linear algebra because they allow us to express any vector in a vector space in terms of a set of basis vectors. This allows us to perform operations such as vector addition, scalar multiplication, and matrix multiplication in a more efficient way. Free bases are also used in solving systems of linear equations and finding the inverse of a matrix.