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Section 2.9 Homogeneous Linear Systems (VS9)

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Subsection 2.9.1 Class Activities

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Definition 2.9.1.

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]
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Activity 2.9.2.

Note that if [a1⋮an] and [b1⋮bn] are solutions to x1v→1+⋯+xnv→n=0→ so is [a1+b1⋮an+bn], since

a1v→1+⋯+anv→n=0→ and b1v→1+⋯+bnv→n=0→

implies

(a1+b1)v→1+⋯+(an+bn)v→n=0→.

Similarly, if c∈R, [ca1⋮can] is a solution. Thus the solution set of a homogeneous system is...

  1. A basis for Rn.

  2. A subspace of Rn.

  3. The empty set.

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Activity 2.9.3.

Consider the homogeneous system of equations

x1+2x2+x4=02x1+4x2−x3−2x4=03x1+6x2−x3−x4=0
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(a)

Find its solution set (a subspace of R4).

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(b)

Rewrite this solution space in the form

{a[????]+b[????]|a,b∈R}.
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(c)

Rewrite this solution space in the form

span{[????],[????]}.
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Activity 2.9.5.

Consider the homogeneous system of equations

2x1+4x2+2x3−4x4=0−2x1−4x2+x3+x4=03x1+6x2−x3−4x4=0

Find a basis for its solution space.
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Activity 2.9.6.

Consider the homogeneous vector equation

x1[2−23]+x2[4−46]+x3[21−1]+x4[−41−4]=[000]

Find a basis for its solution space.
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Activity 2.9.7.

Consider the homogeneous system of equations

x1−3x2+2x3=02x1+6x2+4x3=0x1+6x2−4x3=0

Find a basis for its solution space.
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Observation 2.9.8.

The basis of the trivial vector space is the empty set. You can denote this as either ∅ or {}.

Thus, if 0→ is the only solution of a homogeneous system, the basis of the solution space is ∅.

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Subsection 2.9.2 Videos

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Figure 22. Video: Polynomial and matrix calculations
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Subsection 2.9.3 Slideshow

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Slideshow of activities available at https://teambasedinquirylearning.github.io/linear-algebra/2022/VS9.slides.html.

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Exercises 2.9.4 Exercises

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Exercises available at https://teambasedinquirylearning.github.io/linear-algebra/2022/exercises/#/bank/VS9/.

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Subsection 2.9.5 Mathematical Writing Explorations

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Exploration 2.9.9.

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→=[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 2.9.6 Sample Problem and Solution

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Sample problem Example B.1.13.

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