Are two vector spaces equal if they have the same basis?

The answer is yes.

Is the span of two vectors a subspace?

In mathematics, the linear span (also called the linear hull or just span) of a set S of vectors (from a vector space), denoted span(S), is the smallest linear subspace that contains the set.

Is span the same as subspace?

I know that the span of set S is basically the set of all the linear combinations of the vectors in S. The subspace of the set S is the set of all the vectors in S that are closed under addition and multiplication (and the zero vector).

Can two subspaces have the same orthogonal complement?

The general idea here is that if two subspaces are orthogonal, but they are not orthogonal complements of each other, then there must be other vectors in the space orthogonal to both. Thus their orthogonal complements will overlap a bit.

Are zero vectors linearly independent?

A basis must be linearly independent; as seen in part (a), a set containing the zero vector is not linearly independent.

Can two vector spaces share the same dimension?

No two vector spaces can share the same dimension. If V is a vector space with dim(V) = 6 and S is a subspace with V with dim(S) = 6, then S = V. A set of vectors V in vector space V can be linearly independent or can span V, but cannot do both.

How do you know if two vector spaces are equal?

To show that two finite-dimensional vector spaces are equal, one often uses the following criterion: if V is a finite-dimensional vector space and W is a linear subspace of V with dim(W) = dim(V), then W = V. Rn has the standard basis {e1., en}, where ei is the i-th column of the corresponding identity matrix.

What is span in vector space?

The span of a set of vectors is the set of all linear combinations of the vectors. For example, if and. then the span of v1 and v2 is the set of all vectors of the form sv1+tv2 for some scalars s and t.

Is the zero vector always in a span?

Yes. Depending on your definition of span, it is either the smallest subspace containing a set of vectors (and hence 0 belongs to it because 0 is a member of any subspace) or it is the set of all linear combinations in which case the empty sum convention kicks in.