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Linear Algebra for Team-Based Inquiry Learning: 2023 Edition

Steven Clontz, Drew Lewis

Section 2.7 Homogeneous Linear Systems (EV7)

Subsection 2.7.1 Class Activities

Definition 2.7.1.

A homogeneous system of linear equations is one of the form:
\begin{alignat*}{5} a_{11}x_1 &\,+\,& a_{12}x_2 &\,+\,& \dots &\,+\,& a_{1n}x_n &\,=\,& 0 \\ a_{21}x_1 &\,+\,& a_{22}x_2 &\,+\,& \dots &\,+\,& a_{2n}x_n &\,=\,& 0 \\ \vdots& &\vdots& && &\vdots&&\vdots\\ a_{m1}x_1 &\,+\,& a_{m2}x_2 &\,+\,& \dots &\,+\,& a_{mn}x_n &\,=\,& 0 \end{alignat*}
This system is equivalent to the vector equation:
\begin{equation*} x_1 \vec{v}_1 + \cdots+x_n \vec{v}_n = \vec{0} \end{equation*}
and the augmented matrix:
\begin{equation*} \left[\begin{array}{cccc|c} a_{11} & a_{12} & \cdots & a_{1n} & 0\\ a_{21} & a_{22} & \cdots & a_{2n} & 0\\ \vdots & \vdots & \ddots & \vdots & \vdots\\ a_{m1} & a_{m2} & \cdots & a_{mn} & 0 \end{array}\right] \end{equation*}

Activity 2.7.2.

Consider the homogeneous vector equation \(x_1 \vec{v}_1 + \cdots+x_n \vec{v}_n = \vec{0}\text{.}\)
(a)
Note that if \(\left[\begin{array}{c} a_1 \\ \vdots \\ a_n \end{array}\right] \) and \(\left[\begin{array}{c} b_1 \\ \vdots \\ b_n \end{array}\right] \) are both solutions, we know that
\begin{equation*} a_1 \vec{v}_1+\cdots+a_n \vec{v}_n = \vec{0} \text{ and } b_1 \vec{v}_1+\cdots+b_n \vec{v}_n = \vec{0} \text{.} \end{equation*}
Therefore by adding these equations,
\begin{equation*} (a_1 + b_1) \vec{v}_1+\cdots+(a_n+b_n) \vec{v}_n = \vec{0} \end{equation*}
shows that \(\left[\begin{array}{c} a_1+ b_1 \\ \vdots \\ a_n+b_n \end{array}\right] \) is also a solution. Thus the solution set of a homogeneous system is...
  1. Closed under addition.
  2. Not closed under addition.
  3. Linearly dependent.
  4. Linearly independent.
(b)
Similarly, if \(c \in \IR\text{,}\) \(\left[\begin{array}{c} ca_1 \\ \vdots \\ ca_n \end{array}\right] \) is a solution. Thus the solution set of a homogeneous system is also closed under scalar multiplication, and therefore...
  1. A basis for \(\IR^n\text{.}\)
  2. A subspace of \(\IR^n\text{.}\)
  3. All of \(\IR^n\text{.}\)
  4. The empty set.

Activity 2.7.3.

Consider the homogeneous system of equations
\begin{alignat*}{5} x_1&\,+\,&2x_2&\,\,& &\,+\,& x_4 &=& 0\\ 2x_1&\,+\,&4x_2&\,-\,&x_3 &\,-\,&2 x_4 &=& 0\\ 3x_1&\,+\,&6x_2&\,-\,&x_3 &\,-\,& x_4 &=& 0 \end{alignat*}
(a)
Find its solution set (a subspace of \(\IR^4\)).
(b)
Rewrite this solution space in the form
\begin{equation*} \setBuilder{ a \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\end{array}\right] + b \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown \end{array}\right] }{a,b \in \IR}. \end{equation*}
(c)
Rewrite this solution space in the form
\begin{equation*} \vspan\left\{\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\end{array}\right], \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown \end{array}\right]\right\}. \end{equation*}
(d)
Which of these choices best describes the set of two vectors \(\left\{\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\end{array}\right], \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown \end{array}\right]\right\}\) used in this span?
  1. The set is linearly dependent.
  2. The set is linearly independent.
  3. The set spans all of \(\IR^4\text{.}\)
  4. The set fails to span the solution space.

Activity 2.7.5.

Consider the homogeneous system of equations
\begin{alignat*}{5} 2x_1&\,+\,&4x_2&\,+\,& 2x_3&\,-\,&4x_4 &=& 0 \\ -2x_1&\,-\,&4x_2&\,+\,&x_3 &\,+\,& x_4 &=& 0\\ 3x_1&\,+\,&6x_2&\,-\,&x_3 &\,-\,&4 x_4 &=& 0 \end{alignat*}
Find a basis for its solution space.

Activity 2.7.6.

Consider the homogeneous vector equation
\begin{equation*} x_1 \left[\begin{array}{c} 2 \\ -2 \\ 3 \end{array}\right]+ x_2 \left[\begin{array}{c} 4 \\ -4 \\ 6 \end{array}\right]+ x_3 \left[\begin{array}{c} 2 \\ 1 \\ -1 \end{array}\right]+ x_4 \left[\begin{array}{c} -4 \\ 1 \\ -4 \end{array}\right]= \left[\begin{array}{c} 0 \\ 0 \\ 0 \end{array}\right] \end{equation*}
Find a basis for its solution space.

Activity 2.7.7.

Consider the homogeneous system of equations
\begin{alignat*}{5} x_1&\,-\,&3x_2&\,+\,& 2x_3 &=& 0\\ 2x_1&\,+\,&6x_2&\,+\,&4x_3 &=& 0\\ x_1&\,+\,&6x_2&\,-\,&4x_3 &=& 0 \end{alignat*}
(a)
Find its solution space.
(b)
Which of these is the best choice of basis for this solution space?
  1. \(\displaystyle \{\}\)
  2. \(\displaystyle \{\vec 0\}\)
  3. The basis does not exist

Activity 2.7.8.

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.
(a)
What homoegeneous linear system describes the positions within the level that would be aligned with the pixel \((0,0)\) on the screen?
(b)
Solve this system to describe these locations.

Subsection 2.7.2 Videos

Figure 18. Video: Polynomial and matrix calculations

Subsection 2.7.3 Slideshow

Exercises 2.7.4 Exercises

Subsection 2.7.5 Mathematical Writing Explorations

Exploration 2.7.9.

An \(n \times n\) 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 \(\vec{b} = \left[\begin{array}{c}b_1\\b_2\\ \vdots \\b_n \end{array}\right]\text{,}\) the system of equations given by \(\left[\begin{array}{c|c}M & \vec{b}\end{array} \right]\) has a unique solution.
  • Prove that the columns of \(M\) form a basis for \(\mathbb{R}^n\text{.}\)
  • Prove that the rank of \(M\) is \(n\text{.}\)

Subsection 2.7.6 Sample Problem and Solution

Sample problem Example B.1.11.
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