A vector space in linear algebra is a mathematical structure that consists of a set of objects called vectors and a set of operations that can be performed on these vectors. These operations include vector addition and scalar multiplication, and they must adhere to certain properties for the set to be considered a vector space. Examples of vector spaces include the set of all 2-dimensional vectors, the set of all polynomials of degree n, and the set of all functions from one set to another.
The basic properties of a vector space include closure, commutativity, associativity, existence of a zero vector, existence of additive inverse, existence of a multiplicative identity, and distributivity. Closure means that the result of any operation on vectors in the space is also a vector in the space. Commutativity means that the order of operands does not affect the result of an operation. Associativity means that the grouping of operands does not affect the result of an operation. The existence of a zero vector and additive inverse means that every vector has an opposite or additive inverse that when added to the vector results in the zero vector. The existence of a multiplicative identity means that there is a vector that when multiplied by any vector results in the original vector. Distributivity means that scalar multiplication can be distributed over vector addition.
A vector space consists of vectors, whereas a matrix space consists of matrices. Vectors are 1-dimensional objects, while matrices are 2-dimensional objects. In a vector space, operations like vector addition and scalar multiplication are defined, whereas in a matrix space, matrix addition and matrix multiplication are defined. Additionally, a vector space has a set of properties that must be satisfied, while a matrix space may have different properties depending on the type of matrices in the space.
The dimension of a vector space is the number of vectors in a basis for that space. A basis is a set of linearly independent vectors that span the entire space. For example, the dimension of the set of all 2-dimensional vectors is 2, as any 2 linearly independent vectors can serve as a basis for this space.
In a vector space, linear independence refers to a set of vectors that cannot be written as a linear combination of other vectors in that space. In other words, no vector in a linearly independent set can be expressed as a linear combination of other vectors in that set. This concept is important in determining the dimension of a vector space and in solving systems of linear equations.
