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Section3.1Solution Sets

Objectives
  1. Understand the relationship between the solution set of Ax = 0 and the solution set of Ax = b .
  2. Understand the difference between the solution set and the column span.
  3. Recipes: parametric vector form, write the solution set of a homogeneous system as a span.
  4. Pictures: solution set of a homogeneous system, solution set of an inhomogeneous system, the relationship between the two.
  5. Vocabulary: homogeneous/inhomogeneous, trivial solution.

In this section we will study the geometry of the solution set of any matrix equation Ax = b .

Subsection3.1.1Homogeneous Systems

The equation Ax = b is easier to solve when b = 0, so we start with this case.

Definition

A system of linear equations of the form Ax = 0 is called homogeneous.

A system of linear equations of the form Ax = b for b B = 0 is called inhomogeneous.

A homogeneous system is just a system of linear equations where all constants on the right side of the equals sign are zero.

A homogeneous system always has the solution x = 0. This is called the trivial solution. Any nonzero solution is called nontrivial.

Observation

The equation Ax = 0 has a nontrivial solution ⇐⇒ there is a free variable ⇐⇒ A has a column without a pivot position.

Observation

When we row reduce the augmented matrix for a homogeneous system of linear equations, the last column will be zero throughout the row reduction process. We saw this in the last example:

C 134 02 12 0101 0 D

So it is not really necessary to write augmented matrices when solving homogeneous systems.

When the homogeneous equation Ax = 0 does have nontrivial solutions, it turns out that the solution set can be conveniently expressed as a span.

Parametric Vector Form (homogeneous case)

Consider the following matrix in reduced row echelon form:

A = C 10 8 701430000 D .

The matrix equation Ax = 0 corresponds to the system of equations

T x 1 8 x 3 7 x 4 = 0 x 2 + 4 x 3 + 3 x 4 = 0.

We can write the parametric form as follows:

GMKMI x 1 = 8 x 3 + 7 x 4 x 2 = 4 x 3 3 x 4 x 3 = x 3 x 4 = x 4 .

We wrote the redundant equations x 3 = x 3 and x 4 = x 4 in order to turn the above system into a vector equation:

x = EPN x 1 x 2 x 3 x 4 FQO = x 3 EPN 8 410 FQO + x 4 EPN 7 301 FQO .

This vector equation is called the parametric vector form of the solution set. Since x 3 and x 4 are allowed to be anything, this says that the solution set is the set of all linear combinations of EPN 8 410 FQO and EPN 7 301 FQO . In other words, the solution set is

Span GMKMIEPN 8 410 FQO , EPN 7 301 FQOHMLMJ .

Here is the general procedure.

Recipe: Parametric vector form (homogeneous case)

Let A be an m × n matrix. Suppose that the free variables in the homogeneous equation Ax = 0 are, for example, x 3 , x 6 , and x 8 .

  1. Find the reduced row echelon form of A .
  2. Write the parametric form of the solution set, including the redundant equations x 3 = x 3 , x 6 = x 6 , x 8 = x 8 . Put equations for all of the x i in order.
  3. Make a single vector equation from these equations by making the coefficients of x 3 , x 6 , and x 8 into vectors v 3 , v 6 , and v 8 , respectively.

The solutions to Ax = 0 will then be expressed in the form

x = x 3 v 3 + x 6 v 6 + x 8 v 8

for some vectors v 3 , v 6 , v 8 in R n , and any scalars x 3 , x 6 , x 8 . This is called the parametric vector form of the solution.

In this case, the solution set can be written as Span { v 3 , v 6 , v 8 } .

We emphasize the following fact in particular.

The set of solutions to a homogeneous equation Ax = 0 is a span.

Since there were two variables in the above example, the solution set is a subset of R 2 . Since one of the variables was free, the solution set is a line:

Ax = 0

In order to actually find a nontrivial solution to Ax = 0 in the above example, it suffices to substitute any nonzero value for the free variable x 2 . For instance, taking x 2 = 1 gives the nontrivial solution x = 1 · A 31 B = A 31 B . Compare to this important note in Section 2.3.

Since there were three variables in the above example, the solution set is a subset of R 3 . Since two of the variables were free, the solution set is a plane.

There is a natural question to ask here: is it possible to write the solution to a homogeneous matrix equation using fewer vectors than the one given in the above recipe? We will see in example in Section 3.2 that the answer is no: the vectors from the recipe are always linearly independent, which means that there is no way to write the solution with fewer vectors.

Another natural question is: are the solution sets for inhomogeneuous equations also spans? As we will see shortly, they are never spans, but they are closely related to spans.

There is a natural relationship between the number of free variables and the “size” of the solution set, as follows.

Dimension of the solution set

The above examples show us the following pattern: when there is one free variable in a consistent matrix equation, the solution set is a line, and when there are two free variables, the solution set is a plane, etc. The number of free variables is called the dimension of the solution set.

We will develop a rigorous definition of dimension in Section 3.4, but for now the dimension will simply mean the number of free variables. Compare with this important note in Section 3.2.

Intuitively, the dimension of a solution set is the number of parameters you need to describe a point in the solution set. For a line only one parameter is needed, and for a plane two parameters are needed. This is similar to how the location of a building on Peachtree Street—which is like a line—is determined by one number and how a street corner in Manhattan—which is like a plane—is specified by two numbers.

Subsection3.1.2Inhomogeneous Systems

Recall that a matrix equation Ax = b is called inhomogeneous when b B = 0.

In the above example, the solution set was all vectors of the form

x = R x 1 x 2 S = x 2 R 31 S + R 30 S

where x 2 is any scalar. The vector p = A 30 B is also a solution of Ax = b : take x 2 = 0. We call p a particular solution.

In the solution set, x 2 is allowed to be anything, and so the solution set is obtained as follows: we take all scalar multiples of A 31 B and then add the particular solution p = A 30 B to each of these scalar multiples. Geometrically, this is accomplished by first drawing the span of A 31 B , which is a line through the origin (and, not coincidentally, the solution to Ax = 0 ), and we translate, or push, this line along p = A 30 B . The translated line contains p and is parallel to Span { A 31 B } : it is a translate of a line.

Ax = 0 Ax = b p

In the above example, the solution set was all vectors of the form

x = C x 1 x 2 x 3 D = x 2 C 110 D + x 3 C 201 D + C 100 D .

where x 2 and x 3 are any scalars. In this case, a particular solution is p = C 100 D .

In the previous example and the example before it, the parametric vector form of the solution set of Ax = b was exactly the same as the parametric vector form of the solution set of Ax = 0 (from this example and this example, respectively), plus a particular solution.

Key Observation

If Ax = b is consistent, the set of solutions to is obtained by taking one particular solution p of Ax = b , and adding all solutions of Ax = 0.

In particular, if Ax = b is consistent, the solution set is a translate of a span.

The parametric vector form of the solutions of Ax = b is just the parametric vector form of the solutions of Ax = 0, plus a particular solution p .

It is not hard to see why the key observation is true. If p is a particular solution, then Ap = b , and if x is a solution to the homogeneous equation Ax = 0, then

A ( x + p )= Ax + Ap = 0 + b = b ,

so x + p is another solution of Ax = b . On the other hand, if we start with any solution x to Ax = b then x p is a solution to Ax = 0 since

A ( x p )= Ax Ap = b b = 0.

See the interactive figures in the next subsection for visualizations of the key observation.

Dimension of the solution set

As in this important note, when there is one free variable in a consistent matrix equation, the solution set is a line—this line does not pass through the origin when the system is inhomogeneous—when there are two free variables, the solution set is a plane (again not through the origin when the system is inhomogeneous), etc.

Again compare with this important note in Section 3.2.

Subsection3.1.3Solution Sets and Column Spans

To every m × n matrix A , we have now associated two completely different geometric objects, both described using spans.

  • The solution set: for fixed b , this is the set of all x such that Ax = b .

    • This is a span if b = 0, and it is a translate of a span if b B = 0 (and Ax = b is consistent).
    • It is a subset of R n .
    • It is computed by solving a system of equations: usually by row reducing and finding the parametric vector form.
  • The span of the columns of A : this is the set of all b such that Ax = b is consistent.

    • This is always a span.
    • It is a subset of R m .
    • It is not computed by solving a system of equations: row reduction plays no role.

Do not confuse these two geometric constructions! In the first the question is which x ’s work for a given b and in the second the question is which b ’s work for some x .