Skip to main content
Logo image

Preview of Team-Based Inquiry Learning Library

TBIL Community

Section 19.7 Homogeneous Linear Systems (EV7)

Subsection 19.7.1 Warmup

Remark 19.100.

Recall from Section 19.3 that 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 19.101.

(a)
In Section 19.3, we observed that if
\begin{equation*} x_1 \vec{v}_1 + \cdots+x_n \vec{v}_n = \vec{0} \end{equation*}
is a homogeneous vector equation, then:
  • The zero vector \(\vec{0}\) 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 \(\IR^n\text{.}\)
  2. A subspace of \(\IR^n\text{.}\)
  3. All of \(\IR^n\text{.}\)
  4. The empty set.

Subsection 19.7.2 Class Activities

Activity 19.102.

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)
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 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 19.103.

Consider the homogeneous system of equations
\begin{alignat*}{8} 2x_1&\,+\,&4x_2&\,+\,&2x_3 &\,-\,&3 x_4 &\,+\,&31x_5&\,+\,&2x_6&\,-\,&16x_7&=& 0\\ -1x_1&\,-\,&2x_2&\,+\,&4x_3 &\,-\,&x_4 &\,+\,&2x_5&\,+\,&9x_6&\,+\,&3x_7&=& 0\\ x_1&\,+\,&2x_2&\,+\,&x_3 &\,+\,& x_4 &\,+\,&3x_5&\,+\,&6x_6&\,+\,&7x_7&=& 0 \end{alignat*}
(a)
Find its solution set (a subspace of \(\IR^7\)).
(b)
Rewrite this solution space in the form
\begin{equation*} \setBuilder{ a \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right] + b \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right]+c \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right]+d \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right] }{a,b,c,d \in \IR}. \end{equation*}
(c)
Which of these choices best describes the set of vectors \(\left\{\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right], \left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right],\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right],\left[\begin{array}{c} \unknown \\ \unknown \\ \unknown \\ \unknown\\ \unknown\\ \unknown \\ \unknown\end{array}\right]\right\}\) 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 19.105.

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
Answer.
A.

Activity 19.106.

To create a computer-animated film, an animator first models a scene as a subset of \(\mathbb R^3\text{.}\) 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 \((x,y,z)\in\mathbb R^3\) onto the pixel located at \((x+y,y-z)\in\mathbb R^2\text{.}\)
(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?
(b)
Solve this system to describe these locations.

Subsection 19.7.3 Individual Practice

Activity 19.107.

Let \(S=\setList{ \left[\begin{array}{c} -2 \\ 1 \\ 0 \\ 0\end{array}\right], \left[\begin{array}{c} -1 \\ 0 \\ -4 \\ 1 \end{array}\right], \left[\begin{array}{c} 1 \\ 0 \\ -2 \\ 3 \end{array}\right] }\) and \(A=\left[\begin{array}{ccc} -2 & -1 &1\\ 1 & 0 &0\\ 0 & -4 &-2\\ 0 & 1 &3 \end{array}\right];\) note that
\begin{equation*} \RREF(A)=\left[\begin{array}{ccc} 1 & 0 &0\\ 0 & 1 &0\\ 0 & 0 &1\\ 0 & 0 &0 \end{array}\right]. \end{equation*}
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.
(b)
The matrix \(A\) does not span \(\IR^4\) because \(\RREF(A)\) has a row of zeroes.
(c)
The set of vectors \(S\) spans.
(d)
The set of vectors \(S\) is a basis.

Subsection 19.7.4 Videos

Figure 263. Video: Polynomial and matrix calculations

Subsection 19.7.5 Exercises

Subsection 19.7.6 Mathematical Writing Explorations

Exploration 19.108.

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 19.7.7 Sample Problem and Solution

Sample problem Example C.11.
PrevTopNext