Thread: Are these subspaces?

For each of the following subsets of the indicated vector space V , decide whether it is a subspace.

1. GLn(F) in V = Mn(F).

2. The set Symn ⊆ V = Mn(F) of symmetric matrices: Symn = {A ∈ Mn(F) | AT = A}.
3. For fixed A ∈ V = Mn(F), the centralizer of A C(A) = {B ∈ Mn(F) | AB = BA}.
4. The set of all functions f ∈ V = C(R) for which f(1) ∈ Q.


I missed this class and am not really sure how to determine if something is a subspace so I'm trying to do some practice. Any help would be great.

Originally Posted by amm345

For each of the following subsets of the indicated vector space V , decide whether it is a subspace.

1. GLn(F) in V = Mn(F).

2. The set Symn ⊆ V = Mn(F) of symmetric matrices: Symn = {A ∈ Mn(F) | AT = A}.
3. For fixed A ∈ V = Mn(F), the centralizer of A C(A) = {B ∈ Mn(F) | AB = BA}.
4. The set of all functions f ∈ V = C(R) for which f(1) ∈ Q.


I missed this class and am not really sure how to determine if something is a subspace so I'm trying to do some practice. Any help would be great.

In every case you've to check whether the given set is non-empty, closed under addition and multiplication by scalar. For example in (1), if we aded two invertible matrices do we always get an invertible matrix? Of course not, but now you find a counterexample.

Tonio