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Linear Algebra for Team-Based Inquiry Learning: PREVIEW Edition — Instructor Version

Steven Clontz, Drew Lewis

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Section 2.7 Homogeneous Linear Systems (EV7)

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Subsection 2.7.1 Warmup

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Remark 2.7.1.

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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
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This system is equivalent to the vector equation:
x1v→1+⋯+xnv→n=0→
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and the augmented matrix:
[a11a12⋯a1n0a21a22⋯a2n0⋮⋮⋱⋮⋮am1am2⋯amn0].
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Activity 2.7.2.

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In Section 2.3, we observed that if
x1v→1+⋯+xnv→n=0→
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is a homogenous 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.
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Based on this recollection, which of the following best describes the solution set to the homogenous equation?
  1. A basis for Rn.
  2. A subspace of Rn.
  3. All of Rn.
  4. The empty set.
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Subsection 2.7.2 Class Activities

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Activity 2.7.3.

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Consider the homogeneous system of equations
x1+2x2+x4=02x1+4x2−x3−2x4=03x1+6x2−x3−x4=0
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(a)
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Find its solution set (a subspace of R4).
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(b)
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Rewrite this solution space in the form
{a[????]+b[????]|a,b∈R}.
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(c)
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Rewrite this solution space in the form
span{[????],[????]}.
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(d)
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Which of these choices best describes the set of two vectors {[????],[????]} used in this span?
  1. The set is linearly dependent.
  2. The set is linearly independent.
  3. The set spans all of R4.
  4. The set fails to span the solution space.
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Activity 2.7.5.

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Consider the homogeneous system of equations
2x1+4x2+2x3−4x4=0−2x1−4x2+x3+x4=03x1+6x2−x3−4x4=0
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Find a basis for its solution space.
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Activity 2.7.6.

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Consider the homogeneous vector equation
x1[2−23]+x2[4−46]+x3[21−1]+x4[−41−4]=[000]
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Find a basis for its solution space.
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Activity 2.7.7.

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Consider the homogeneous system of equations
x1−3x2+2x3=02x1+6x2+4x3=0x1+6x2−4x3=0
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(b)
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Which of these is the best choice of basis for this solution space?
  1. {}
  2. {0→}
  3. The basis does not exist
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Activity 2.7.8.

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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 tranformation that maps visual information at the point (x,y,z)∈R3 onto the pixel located at (x+y,y−z)∈R2.
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(a)
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What homoegeneous linear system describes the positions (x,y,z) within the original scene that would be aligned with the pixel (0,0) on the screen?
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(b)
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Solve this system to describe these locations.
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Subsection 2.7.3 Individual Practice

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Activity 2.7.9.

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Let S={[−2100],[−10−41],[10−23]} and A=[−2−111000−4−2013]; note that
RREF(A)=[100010001000].
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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.
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(a)
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The matrix A is linearly independent because RREF(A) has a pivot in each column.
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(b)
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The matrix A does not span R4 because RREF(A) has a row of zeroes.
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(d)
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The set of vectors S is a basis.
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Subsection 2.7.4 Videos

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Figure 18. Video: Polynomial and matrix calculations
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Exercises 2.7.5 Exercises

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Subsection 2.7.6 Mathematical Writing Explorations

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Exploration 2.7.10.

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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→=[b1b2⋮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.
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Subsection 2.7.7 Sample Problem and Solution

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Sample problem Example B.1.11.
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