## Section2.9Homogeneous Linear Systems (VS9)

### Subsection2.9.1Class Activities

#### Definition2.9.1.

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*}

#### Activity2.9.2.

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 solutions to $$x_1 \vec{v}_1 + \cdots+x_n \vec{v}_n = \vec{0}$$ so is $$\left[\begin{array}{c} a_1 +b_1\\ \vdots \\ a_n+b_n \end{array}\right] \text{,}$$ since

\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} \end{equation*}

implies

\begin{equation*} (a_1 + b_1) \vec{v}_1+\cdots+(a_n+b_n) \vec{v}_n = \vec{0} . \end{equation*}

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...

1. A basis for $$\IR^n\text{.}$$

2. A subspace of $$\IR^n\text{.}$$

3. The empty set.

#### Activity2.9.3.

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*}

#### Activity2.9.5.

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.

#### Activity2.9.6.

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.

#### Activity2.9.7.

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*}

Find a basis for its solution space.

#### Observation2.9.8.

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

Thus, if $$\vec{0}$$ is the only solution of a homogeneous system, the basis of the solution space is $$\emptyset\text{.}$$

### Subsection2.9.3Slideshow

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

### Subsection2.9.5Mathematical Writing Explorations

#### Exploration2.9.9.

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 $$n\text{.}$$

### Subsection2.9.6Sample Problem and Solution

Sample problem Example B.1.13.