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Section3.3Subspaces

Objectives
  1. Learn the definition of a subspace.
  2. Learn to determine whether or not a subset is a subspace.
  3. Learn the most important examples of subspaces.
  4. Learn to write a given subspace as a column space or null space.
  5. Recipe: compute a spanning set for a null space.
  6. Picture: whether a subset of R 2 or R 3 is a subspace or not.
  7. Vocabulary: subspace, column space, null space.

In this section we discuss subspaces of R n . A subspace turns out to be exactly the same thing as a span, except we don’t have a particular set of spanning vectors in mind. This change in perspective is quite useful, as it is easy to produce subspaces that are not obviously spans. For example, the solution set of the equation x + 3 y + z = 0 is a span because the equation is homogeneous, but we would have to compute the parametric vector form in order to write it as a span.

x + 3 y + z = 0

(A subspace also turns out to be the same thing as the solution set of a homogeneous system of equations.)

Subsection3.3.1Subspaces: Definition and Examples

Definition

A subset of R n is any collection of points of R n .

For instance, the unit circle

C = C ( x , y ) in R 2 EE x 2 + y 2 = 1 D

is a subset of R 2 .

Above we expressed C in set builder notation: in English, it reads C is the set of all ordered pairs ( x , y ) in R 2 such that x 2 + y 2 = 1.

Definition

A subspace of R n is a subset V of R n satisfying:

  1. Non-emptiness: The zero vector is in V .
  2. Closure under addition: If u and v are in V , then u + v is also in V .
  3. Closure under scalar multiplication: If v is in V and c is in R , then cv is also in V .

As a consequence of these properties, we see:

  • If v is a vector in V , then all scalar multiples of v are in V by the third property. In other words the line through any nonzero vector in V is also contained in V .
  • If u , v are vectors in V and c , d are scalars, then cu , dv are also in V by the third property, so cu + dv is in V by the second property. Therefore, all of Span { u , v } is contained in V
  • Similarly, if v 1 , v 2 ,..., v n are all in V , then Span { v 1 , v 2 ,..., v n } is contained in V . In other words, a subspace contains the span of any vectors in it.

If you choose enough vectors, then eventually their span will fill up V , so we already see that a subspace is a span. See this theorem below for a precise statement.

Example

The set R n is a subspace of itself: indeed, it contains zero, and is closed under addition and scalar multiplication.

Example

The set { 0 } containing only the zero vector is a subspace of R n : it contains zero, and if you add zero to itself or multiply it by a scalar, you always get zero.

Subsets versus Subspaces

A subset of R n is any collection of vectors whatsoever. For instance, the unit circle

C = C ( x , y ) in R 2 EE x 2 + y 2 = 1 D

is a subset of R 2 , but it is not a subspace. In fact, all of the non-examples above are still subsets of R n . A subspace is a subset that happens to satisfy the three additional defining properties.

In order to verify that a subset of R n is in fact a subspace, one has to check the three defining properties. That is, unless the subset has already been verified to be a subspace: see this important note below.

Subsection3.3.2Common Types of Subspaces

Proof

If V = Span { v 1 , v 2 ,..., v p } , we say that V is the subspace spanned by or generated by the vectors v 1 , v 2 ,..., v p . We call { v 1 , v 2 ,..., v p } a spanning set for V .

Any matrix naturally gives rise to two subspaces.

Definition

Let A be an m × n matrix.

  • The column space of A is the subspace of R m spanned by the columns of A . It is written Col ( A ) .
  • The null space of A is the subspace of R n consisting of all solutions of the homogeneous equation Ax = 0:
    Nul ( A )= C x in R n EE Ax = 0 D .

The column space is defined to be a span, so it is a subspace by the above theorem. We need to verify that the null space is really a subspace. In Section 3.1 we already saw that the set of solutions of Ax = 0 is always a span, so the fact that the null spaces is a subspace should not come as a surprise.

The column space and the null space of a matrix are both subspaces, so they are both spans. The column space of a matrix A is defined to be the span of the columns of A . The null space is defined to be the solution set of Ax = 0, so this is a good example of a kind of subspace that we can define without any spanning set in mind. In other words, it is easier to show that the null space is a subspace than to show it is a span—see the proof above. In order to do computations, however, it is usually necessary to find a spanning set.

Null Spaces are Solution Sets

The null space of a matrix is the solution set of a homogeneous system of equations. For example, the null space of the matrix

A = F 172 2134 2 3 G

is the solution set of Ax = 0, i.e., the solution set of the system of equations

H x + 7 y + 2 z = 0 2 x + y + 3 z = 04 x 2 y 3 z = 0.

Conversely, the solution set of any homogeneous system of equations is precisely the null space of the corresponding coefficient matrix.

To find a spanning set for the null space, one has to solve a system of homogeneous equations.

Recipe: Compute a spanning set for a null space

To find a spanning set for Nul ( A ) , compute the parametric vector form of the solutions to the homogeneous equation Ax = 0. The vectors attached to the free variables form a spanning set for Nul ( A ) .

Writing a subspace as a column space or a null space

A subspace can be given to you in many different forms. In practice, computations involving subspaces are much easier if your subspace is the column space or null space of a matrix. The simplest example of such a computation is finding a spanning set: a column space is by definition the span of the columns of a matrix, and we showed above how to compute a spanning set for a null space using parametric vector form. For this reason, it is useful to rewrite a subspace as a column space or a null space before trying to answer questions about it.

When asking questions about a subspace, it is usually best to rewrite the subspace as a column space or a null space.

This also applies to the question “is my subset a subspace?” If your subset is a column space or null space of a matrix, then the answer is yes.

Example

Let

V = KI ab J in R 2 EE 2 a = 3 b L

be the subset of a previous example. The subset V is exactly the solution set of the homogeneous equation 2 x 3 y = 0. Therefore,

V = Nul A 2 3 B .

In particular, it is a subspace. The reduced row echelon form of A 2 3 B is A 1 3 / 2 B , so the parametric form of V is x = 3 / 2 y , so the parametric vector form is A xy B = y A 3 / 21 B , and hence CA 3 / 21 BD spans V .