Linear Algebra Done Right · Problem 3E.11

PART (A)

To show $U$ is a subspace, you need to verify the three closure properties: the zero vector belongs to $U$ (it has zero nonzero coordinates), the sum of any two vectors in $U$ stays in $U$ (finitely many nonzero entries plus finitely many nonzero entries is still finite), and scalar multiples preserve membership. The key insight is that "finitely many nonzero coordinates" is preserved under these operations.

PART (B)

The quotient space $\mathbf{F}^\infty / U$ consists of equivalence classes of infinite sequences. Two sequences are equivalent if their difference lies in $U$—that is, if they differ in only finitely many places. To prove infinite dimensionality, construct an infinite linearly independent set of cosets by choosing sequences that differ in infinitely many positions in a controlled way, so no finite linear combination can collapse them to the zero coset.

PART (A)

To show $U$ is a subspace, you need to verify the three closure properties: the zero vector belongs to $U$ (it has zero nonzero coordinates), the sum of any two vectors in $U$ stays in $U$ (finitely many nonzero entries plus finitely many nonzero entries is still finite), and scalar multiples preserve membership. The key insight is that "finitely many nonzero coordinates" is preserved under these operations.

PART (B)

The quotient space $\mathbf{F}^\infty / U$ consists of equivalence classes of infinite sequences. Two sequences are equivalent if their difference lies in $U$—that is, if they differ in only finitely many places. To prove infinite dimensionality, construct an infinite linearly independent set of cosets by choosing sequences that differ in infinitely many positions in a controlled way, so no finite linear combination can collapse them to the zero coset.