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Section3.6Subspaces

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. Recipe: compute a spanning set for a null space.
  5. Picture: whether a subset of R 2 or R 3 is a subspace or not.
  6. Vocabulary words: 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 which 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 to find a spanning set of vectors.

x + 3 y + z = 0

Subsection3.6.1Subspaces: Definition and Examples

Definition

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

For instance, the unit circle

C = A ( x , y ) in R 2 CC x 2 + y 2 = 1 B

is a subset of R 2 .

Above we expressed C in set builder notation.

Definition

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

  1. Nonemptiness: 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 v is also contained in V .
  • If u , v are vectors in V and x , y are scalars, then xu , yv are also in V by the third property, so xu + yv 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.

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 = A ( x , y ) in R 2 CC x 2 + y 2 = 1 B

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.

Subsection3.6.2Common 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 .

Every subspace is a span, and every span is a subspace.

A 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 set of all solutions of the homogeneous equation Ax = 0:
    Nul ( A )= A x in R n CC Ax = 0 B .
    This is a subspace of R n .

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 a span.

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 order to do computations, however, it is usually necessary to find a spanning set.

To be clear: the null space is the solution set of a (homogeneous) system of equations. For example, the null space of the matrix

A = D 172 2134 2 3 E

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

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

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

How do you know if a subset is a subspace?

It is rarely necessary to verify the three properties of a subspace directly: most of the verifications were already done in this section.

  • Is your subset a span? Can it be written as a span?
  • Can it be written as the column space of a matrix?
  • Can it be written as the null space of a matrix? In other words, is it the solution set of a homogeneous system of equations?
  • Is it all of R n or the zero subspace { 0 } ?
  • Can it be written as a type of subspace that we will see later (eigenspace, orthogonal complement, ...)?

If so, then it is automatically a subspace. If all else fails:

  • Can you verify directly that it satisfies the three defining properties of a subspace?