4 Linear algebra

4.4 Subspaces

When we talk about a vector space over a field 𝔽, the word scalar refers to an element of 𝔽.

Definition 4.4.1.

A subspace of a vector space V is a subset U of V which

  1. 1.

    contains the zero vector 𝟎V,

  2. 2.

    is closed under addition, meaning that for all 𝐯,𝐰U we have 𝐯+𝐰U, and

  3. 3.

    is closed under scalar multiplication, meaning that for all scalars λ and all 𝐮U we have λ𝐮U.

We write UV to mean that U is a subspace of V.

The idea this definition captures is that a subspace of V is a nonempty subset which is itself a vector space under the same addition and scalar multiplication as V.

If UV and 𝐮1,,𝐮nU and λ1,,λn are scalars then i=1nλi𝐮iU. This follows by using closure under scalar multiplication and closure under addition lots of times.

4.4.1 Subspace examples

Example 4.4.1.

If V is any vector space, VV. This is because, as a vector space, V contains the zero vector, is closed under addition, and is closed under scalar multiplication.

A subspace of V other than V is called a proper subspace.

Example 4.4.2.

For any vector space V we have {𝟎V}V. Certainly this set contains the zero vector. It is closed under addition because 𝟎V+𝟎V=𝟎V, and it is closed under scalar multiplication by Lemma 4.3.3. This is called the zero subspace.

Example 4.4.3.

Let U be the set of vectors in 2 whose first entry is zero. Then U2. We check the three conditions in the definition of subspace.

  1. 1.

    The zero vector in 2 is (00). This has first coordinate 0, so it is an element of U.

  2. 2.

    Let 𝐯,𝐰U, so that 𝐯=(0x) and 𝐰=(0y) for some real numbers x and y. Then 𝐯+𝐰=(0x+y) has first coordinate 0, so it is an element of U.

  3. 3.

    Let v be as above and λ. Then λ𝐯=(0λx) which has first coordinate 0, so λ𝐯U.

All three conditions hold, so U2. Of course, a similar argument shows the vectors in 𝔽n with first entry 0 are a subspace of 𝔽n for any field 𝔽 and any n.

To every matrix A we associate two important subspaces. The nullspace N(A) (Definition 3.6.5) is the set of all vectors 𝐱 such that A𝐱=𝟎, and the column space C(A) is the set of all linear combinations of the columns of A.

Example 4.4.4.

Let A be an m×n matrix with entries from the field 𝔽. The nullspace N(A) contains the zero vector as A𝟎n=𝟎m. It is closed under addition as if 𝐮,𝐯N(A) then A𝐯=𝟎m and A𝐮=𝟎m so

A(𝐮+𝐯) =A𝐮+A𝐯
=𝟎m+𝟎m
=𝟎m

and therefore 𝐮+𝐯N(A). It is closed under scalar multiplication because if λ is any scalar then A(λ𝐮)=λA𝐮=λ𝟎m=𝟎m so λ𝐮N(A). Therefore N(A)𝔽n.

The column space C(A), defined to be the set of all linear combinations of the columns of A, is a subspace of 𝔽m. We won’t prove that here, because it is a special case of Proposition 4.7.1 which we prove later.

Example 4.4.5.

The set U of all vectors in 3 with first entry 1 is not a subspace of 3. It doesn’t contain the zero vector (and it doesn’t meet the other two conditions either).

Example 4.4.6.

is not a subspace of . It contains the zero vector 0, it is closed under addition because if you add two integers you get another integer. But it is not closed under scalar multiplication: 2 is a scalar, 1, but 2×1 is not in .

Example 4.4.7.

Let U be the set of all functions f: with f(1)=0. This is a subspace of the vector space of all functions . The zero vector in is the constant function that always takes the value zero, so certainly it belongs to U. If f,gU then (f+g)(1)=f(1)+g(1)=0+0=0, so f+gU. If λ and fU then (λf)(1)=λf(1)=λ×0=0 so λfU.

Example 4.4.8.

{AMn×n():AT=A}Mn×n(). The transpose operation satisfies (A+B)T=AT+BT and (λA)T=λAT, which you should check. This makes the three conditions straightforward to check.

Example 4.4.9.

U={AMn×n():A2=𝟎m×n} is not a subspace of Mn×n(). For example, U contains E12=(0100) and E21=(0010) but you can check that E12+E21U.