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Linear Algebra for Team-Based Inquiry Learning: 2023 Edition

Steven Clontz, Drew Lewis

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Section 2.7 Homogeneous Linear Systems (EV7)

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Subsection 2.7.1 Class Activities

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Definition 2.7.1.

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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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Consider the homogeneous vector equation x1v→1+⋯+xnv→n=0→.
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(a)
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Note that if [a1⋮an] and [b1⋮bn] are both solutions, we know that
a1v→1+⋯+anv→n=0→ and b1v→1+⋯+bnv→n=0→.
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Therefore by adding these equations,
(a1+b1)v→1+⋯+(an+bn)v→n=0→
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shows that [a1+b1⋮an+bn] is also a solution. Thus the solution set of a homogeneous system is...
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    Closed under addition.
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    Not closed under addition.
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    Linearly dependent.
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    Linearly independent.
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(b)
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Similarly, if c∈R, [ca1⋮can] is a solution. Thus the solution set of a homogeneous system is also closed under scalar multiplication, and therefore...
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    A basis for Rn.
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    A subspace of Rn.
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    All of Rn.
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    The empty set.
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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?
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    The set is linearly dependent.
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    The set is linearly independent.
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    The set spans all of R4.
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    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?
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    {0→}
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    The basis does not exist
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Activity 2.7.8.

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Suppose that in a certain 3D video game, the “camera” aligns the position (x,y,z) within the level onto the pixel located at (x+y,y−z) on the television screen.
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(a)
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What homoegeneous linear system describes the positions within the level 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 2.7.2 Videos

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Figure 18. Video: Polynomial and matrix calculations
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Subsection 2.7.3 Slideshow

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Exercises 2.7.4 Exercises

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Subsection 2.7.5 Mathematical Writing Explorations

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Exploration 2.7.9.

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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.
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    Prove that the reduced row echelon form of M is the identity matrix.
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    Prove that, for any column vector b→=[b1b2⋮bn], the system of equations given by [Mb→] has a unique solution.
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    Prove that the columns of M form a basis for Rn.
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    Prove that the rank of M is n.
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Subsection 2.7.6 Sample Problem and Solution

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Sample problem Example B.1.11.
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