A subspace of a vector space is a subset of the vector space that still satisfies the properties of a vector space. This means that it is closed under addition and scalar multiplication, and contains the zero vector.
To determine if a subset is a subspace of a vector space, you must check if it satisfies the three properties of a vector space: closure under addition, closure under scalar multiplication, and containing the zero vector. If all three properties are satisfied, then the subset is a subspace.
Yes, a subspace of a vector space can contain infinite elements. This is because the properties of a vector space do not specify a limit on the number of elements that can be contained in a subset.
A subspace is a subset of a vector space that satisfies the properties of a vector space, while a spanning set is a set of vectors that can be used to represent all the vectors in a vector space. A subspace can be created using a spanning set, but a spanning set does not necessarily create a subspace.
Subspaces and linear independence are closely related. A set of vectors is linearly independent if none of the vectors can be written as a linear combination of the others. A subspace can be created using a linearly independent set of vectors, as they form a basis for the subspace.