### Subspace

##### 1. Definition

One number system may be included in other. For example, the integers Z is contained in the real numbers R. Moreover, the addition and multiplication are compatible: The equalities 1 + 2 = 3, 2×3 = 6 hold either as integer operations or real number operations. Therefore we call Z a sub-number system of the number system R. The same remark applies to other inclusions such as ZQ or RC. The similar situation may happen for one vector space contained in another in a compatible way.

A subset H of a vector space V is a subspace if it is closed under two operations:

• u, vHu + vH.
• uH, cRcuH.

By taking c = 0 in the second property, we see that a subspace must contain the zero vector 0. Moreover, by combining the two properties, a subspace is also closed under linear combinations.

Example In any vector space V, the zero vector form the zero subspace {0}: u, v ∈ {0} ⇒ u = v = 0u + v = 0 ∈ {0}, cu = 0 ∈ {0}.

Example C(R) is a subspace of F(R) because the addition and scalar multiplication of continuous functions are still continuous. More generally, Ck(R) is also a subspace of F(R). On the other hand, if k > k', then Ck'(R) is a subspace of Ck(R).

The subset H = {fF(R): f(2) = 0} is also a subspace: If f(2) = g(2) = 0, then (f + g)(2) = f(2) + g(2) = 0 and (cf)(2) = c(f(2)) = 0.

The subset K = {fF(R): f(t) = f(-t)} of even functions is also a subspace: If f(t) = f(-t) and g(t) = g(-t), then (f + g)(t) = f(t) + g(t) = f(-t) + g(-t) = (f + g)(-t) and (cf)(t) = c(f(t)) = c(f(-t)) = (cf)(-t).

On the other hand, the subset H' = {fF(R): f(2) = 1} is not a subspace of F(R) because it does not contain the zero function, which is the zero vector of the vector space F(R).

Example P3 is a subspace of P5. On the other hand, H = {polynomials of degree ≥ 3 and ≤ 5} is not a subspace of P5. A counterexample is given by p(t) = 1 + t4 and q(t) = - t4. We have p, qH but (p + q)(t) = 1 ∉ H.

Example The solutions H1 = {(x, y, z): x + y + z = 0} of a homogeneous equation is a subspace of R3: (x, y, z), (x', y', z') ∈ H1x + y + z = x' + y' + z' = 0 ⇒ (x + x') + (y + y') + (z + z') = 0 and cx + cy + cz = 0.

The solutions H2 = {(x, y, z): x + y + z = 1} of a non-homogeneous equation is not a subspace of R3 because it does not contain the zero vector (0, 0, 0).

The first quadrant H3 = {(x, y): x ≥ 0, y ≥ 0} of R2 is not a subspace because u = (1, 1) ∈ H3 but (-1)u = - u = (-1, -1) ∉ H3. The following shows more subsets of R2 that are not subspaces.

The following shows that a (infinite) straight line passing through the origin is a subspace, while the one not passing through the origin is not a subspace.

Example R2 is not a subspace of R3. In fact, R2 is defined as pairs of numbers, while R3 is triples of numbers, so that R2 is not contained in R3.

On the other hand, R2 can be embedded into R3 as (but not is) subspaces by maps such as (x, y) ⇒ (x, y, 0). This is perhaps where our intuition that R2 is somehow contained in R3 comes from. However, the intuition is ambiguous because of the existence of other embeddings, such as (x, y) ⇒ (0, x, y) or (x, y) ⇒ (x, x, y). Therefore, to be on the safe side, it is better not to confuse R2 with (2-dimensional) subspaces of R3.

Example The set Sym(n) of symmetric matrices is a subspace of the vector space M(n, n) of n by n matrices.

In our examples of subspaces, we saw some appeared also as examples of vector spaces. Indeed we have

A subspace, with the inherited operations, is a vector space.

The proof is a matter of checking the eight properties. In fact, most of the properties are inherited from the bigger space. For example, since u + v = v + u for vectors in V, the equality also holds when restricted to vectors in H. The only things that need further explanations are: For the third property, we already know 0H. For the fourth property, the equality - u = (-1)u shows that uH ⇒ - uH.