Exam 7: Vector Spaces

If S is a linearly independent subset of a vector space V, then S spans V.

If and are infinite-dimensional subspaces of a vector space V, then is an infinite-dimensional subspace of V.

Determine if is in the subspace of given by .

Find a basis for the subspace S of consisting of all linear transformations such that for some diagonal matrix D.

If every infinite set S in a vector space V is linearly dependent, then .

Determine if the set of vectors is linearly independent in .

If S is a linearly independent subset of a vector space V, but S does not span V, then there exists a vector v in V such that the set is also linearly independent.

Determine if is in the subspace of given by .

Determine if the given set is a basis for the vector space .

In the vector space , let S be the set of matrices A such that A is not invertible. Determine if S is a subspace of , and if not explain why.

Determine if is in the subspace of given by .

Every vector space V has a finite set S which both spans V and is linearly independent in V.

Determine the dimension of the subspace S of consisting of all symmetric matrices.

Determine if the set is a basis for the vector space .

Find a basis for the subspace S of consisting of all polynomials p satisfying .

In the vector space , let S be the set of all sequences such that there exists some such that for all (all eventually-zero sequences). Determine if S is a subspace of , and if not explain why.

Determine if the set of vectors spans P.

Determine the dimension of the subspace S of consisting of all sequences such that the series is convergent and satisfies .

Find a basis for the subspace S of P consisting of all even polynomials, that is, all polynomials p satisfying for all x.

Determine if the set of vectors is linearly independent in .