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Section3.4Solution 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 words: homogeneous/inhomogeneous, trivial solution.

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

Subsection3.4.1Homogeneous 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 A = 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.

Observation

In the above example, the last column of the augmented matrix

E 134 02 12 0101 0 F

will be zero throughout the row reduction process. So it is not really necessary to write augmented matrices when solving homogeneous systems.

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.

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 .

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.

Since there were four variables in the above example, the solution set is a subset of R 4 . Since two of the variables were free, the solution set is a plane.

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 x i , x j , x k ,.... Then the solutions to Ax = 0 can be written in the form

x = x i v i + x j v j + x k v k + ···

for some vectors v i , v j , v k ,... in R n , and any scalars x i , x j , x k ,.... This is called the parametric vector form of the solution. It is obtained by finding the parametric form of the solution, including the redundant equations x i = x i , x j = x j , x k = x k ,..., putting these equations in order, and making a vector equation.

In this case, the solution set can be written Span C v i , v j , v k ,... D .

Note that the solution set of a homogeneous equation Ax = 0 is a span!

Subsection3.4.2Inhomogeneous Systems

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

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

x = G x 1 x 2 H = x 2 G 31 H + G 30 H

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

x = E x 1 x 2 x 3 F = x 2 E 110 F + x 3 E 201 F + E 100 F .

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

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

The set of solutions to Ax = b , if it is nonempty, is obtained by taking one particular solution p of Ax = b , and adding all solutions of Ax = 0.

In particular, the solution set of Ax = b is either empty, or it 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 .

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

Dimension of the solution set

As in the first subsection, 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, etc.

We will develop a rigorous definition of dimension in Section 3.7, but for now it is important to note that the “dimension” of the solution set of a consistent system is equal to the number of free variables.

Subsection3.4.3Solution 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 (or it is empty) if b A = 0.
    • 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!