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Section 2.7 Homogeneous Linear Systems (EV7)

Subsection 2.7.1 Warmup

Remark 2.7.1.

Recall from Section 2.3 that a homogeneous system of linear equations is one of the form:
a11x1+a12x2+…+a1nxn=0a21x1+a22x2+…+a2nxn=0⋮⋮⋮⋮am1x1+am2x2+…+amnxn=0
This system is equivalent to the vector equation:
x1v→1+⋯+xnv→n=0→
and the augmented matrix:
[a11a12⋯a1n0a21a22⋯a2n0⋮⋮⋱⋮⋮am1am2⋯amn0].

Activity 2.7.2.

(a)
In Section 2.3, we observed that if
x1v→1+⋯+xnv→n=0→
is a homogeneous vector equation, then:
  • The zero vector 0→ is a solution;
  • The sum of any two solutions is again a solution;
  • Multiplying a solution by a scalar produces another solution.
(b)
Based on this recollection, which of the following best describes the solution set to the homogeneous equation?
  1. A basis for Rn.
  2. A subspace of Rn.
  3. All of Rn.
  4. The empty set.

Subsection 2.7.2 Class Activities

Activity 2.7.3.

Consider the homogeneous system of equations
x1+2x2+x4=02x1+4x2−x3−2x4=03x1+6x2−x3−x4=0
(c)
Which of these choices best describes the set of two vectors {[????],[????]} used in this solution space?
  1. The set is linearly dependent.
  2. The set is linearly independent.
  3. The set spans the solution space.
  4. The set is a basis of the solution space.
Answer.
D.

Activity 2.7.4.

Consider the homogeneous system of equations
2x1+4x2+2x3−3x4+31x5+2x6−16x7=0−1x1−2x2+4x3−x4+2x5+9x6+3x7=0x1+2x2+x3+x4+3x5+6x6+7x7=0
(c)
Which of these choices best describes the set of vectors {[???????],[???????],[???????],[???????]} used in this solution space?
  1. The set is linearly dependent.
  2. The set is linearly independent.
  3. The set spans the solution space.
  4. The set is a basis for the solution space.
Answer.
D.

Activity 2.7.7.

To create a computer-animated film, an animator first models a scene as a subset of R3. Then to transform this three-dimensional visual data for display on a two-dimensional movie screen or television set, the computer could apply a linear transformation that maps visual information at the point (x,y,z)∈R3 onto the pixel located at (x+y,y−z)∈R2.
(a)
What homogeneous linear system describes the positions (x,y,z) within the original scene that would be aligned with the pixel (0,0) on the screen?

Subsection 2.7.3 Individual Practice

Activity 2.7.8.

Let S={[−2100],[−10−41],[10−23]} and A=[−2−111000−4−2013]; note that
RREF(A)=[100010001000].
The following statements are all invalid for at least one reason. Determine what makes them invalid and, suggest alternative valid statements that the author may have meant instead.
(a)
The matrix A is linearly independent because RREF(A) has a pivot in each column.

Subsection 2.7.4 Videos

Figure 18. Video: Polynomial and matrix calculations

Subsection 2.7.5 Exercises

Subsection 2.7.6 Mathematical Writing Explorations

Exploration 2.7.9.

An n×n matrix M is non-singular if the associated homogeneous system with coefficient matrix M is consistent with one solution. Assume the matrices in the writing explorations in this section are all non-singular.
  • Prove that the reduced row echelon form of M is the identity matrix.
  • Prove that, for any column vector b→=[b1b2â‹®bn], the system of equations given by [Mb→] has a unique solution.
  • Prove that the columns of M form a basis for Rn.
  • Prove that the rank of M is n.

Subsection 2.7.7 Sample Problem and Solution