A subspace is a subset of a vector space that satisfies the properties of a vector space. This means that it must contain the zero vector, be closed under addition and scalar multiplication, and must satisfy the axioms of a vector space.
To find the basis for a subspace, you need to first determine the vectors that span the subspace. This can be done by setting up a system of equations and solving for the variables. Once you have the spanning vectors, you can determine which ones are linearly independent to form the basis.
The basis for a subspace is important because it allows us to represent any vector in that subspace as a linear combination of the basis vectors. This makes it easier to perform calculations and transformations on vectors within the subspace.
Yes, there can be multiple bases for the same subspace. This is because as long as the basis vectors are linearly independent and span the subspace, they can be rearranged or multiplied by a scalar and still form a basis for the subspace.
To determine if a set of vectors is a basis for a subspace, you can check if they are linearly independent and span the subspace. This can be done by setting up a system of equations and solving for the variables. If the vectors are linearly independent and span the subspace, then they form a basis for that subspace.