A subspace is a subset of a vector space that contains all the elements of the original vector space and satisfies the axioms of a vector space, such as closure under addition and scalar multiplication.
This means that both sets of vectors contain enough elements to create any vector within the subspace. In other words, any vector in the subspace can be written as a linear combination of vectors from either set.
To show that two sets of vectors span the same subspace, you can demonstrate that any vector in the subspace can be written as a linear combination of vectors from both sets. This can be done by solving a system of equations or by showing that all vectors in the subspace can be expressed as a linear combination of basis vectors from both sets.
This indicates that the two sets of vectors are equivalent in terms of their ability to define the subspace. It also suggests that the two sets may have similar properties or relationships between their vectors.
Yes, it is possible for two sets of vectors to span multiple subspaces. This can occur if the two sets contain vectors that are linearly independent and can define different subspaces within the same vector space.