Section 1.3 Homogeneous Equations
Vectors in [latex]\mathbb{R}^2[/latex]
Definition: 1. A matrix with only one column is called a column vector, or
simply a vector.
2. The set of all vectors with 2 entries is denoted by [latex]\mathbb{R}^2[/latex] (read “r-two”).
3. Two vectors in [latex]\mathbb{R}^2[/latex] are equal if and only if their corresponding entries are equal.
4. Given two vectors [latex]\vec{u} = \begin{bmatrix}u_{1}\\u_{2}\end{bmatrix}[/latex] and [latex]\vec{v} = \begin{bmatrix}v_{1}\\v_{2}\end{bmatrix}[/latex] in [latex]\mathbb{R}^2[/latex], their sum, [latex]\vec{u} + \vec{v}[/latex], is the vector obtained by adding corresponding entries of [latex]\vec{u}[/latex] and [latex]\vec{v}[/latex], i.e. [latex]\vec{u}+\vec{v} = \begin{bmatrix}u_{1}+v_{1}\\u_{2}+v_{2}\end{bmatrix}[/latex].
5. Given a vector [latex]\vec{u} = \begin{bmatrix}u_{1}\\u_{2}\end{bmatrix}[/latex] and a real number [latex]c[/latex], the scalar multiple of [latex]\vec{u} = \begin{bmatrix}u_{1}\\u_{2}\end{bmatrix}[/latex] by [latex]c[/latex] is the vector [latex]c\vec{u} = \begin{bmatrix}cu_{1}\\cu_{2}\end{bmatrix}[/latex] obtained by multiplying each entry in [latex]\vec{u}[/latex] by [latex]c[/latex].
NOTE: You MUST write [latex]\vec{u}[/latex] as vector and [latex]u[/latex] as a number.
Definition: 1. Vectors in [latex]\mathbb{R}^n[/latex] are [latex]n \times 1[/latex] column matrices with [latex]n[/latex] entries where [latex]n[/latex] is a positive integer. We write [latex]\vec{u} = \begin{bmatrix}u_{1}\\u_{2}\\\vdots \\u_{n-1}\\u_{n}\end{bmatrix}[/latex]
2. The vector whose entries are all zero is called the zero vector and is
denoted by [latex]\vec{0}[/latex]
Definition: If [latex]\vec{v}_{1}, \cdots, \vec{v}_{p}[/latex] are vectors in [latex]\mathbb{R}^n[/latex], and if [latex]a_{1}, \cdots, a_{p}[/latex] are constants then [latex]a_{1}\vec{v}_{1}+\cdots+a_{p}\vec{v}_{p}[/latex] is a linear combination of vectors [latex]\vec{v}_{1}, \cdots, \vec{v}_{p}[/latex]
Facts/Properties:
- [latex]\vec{u}+\vec{v}=\vec{v}+\vec{u}[/latex]
- [latex](\vec{u}+\vec{v})+\vec{w}=\vec{u}+(\vec{v}+\vec{w})[/latex]
- [latex]\vec{u}+\vec{0}=\vec{0}+\vec{u} =\vec{u}[/latex]
- [latex]\vec{u}+(-\vec{u})=-\vec{u}+\vec{u}=\vec{0}[/latex]
- [latex]c(\vec{u}+\vec{v})=c\vec{u}+c\vec{v}[/latex]
- [latex](c+d)\vec{u}=c\vec{u}+d\vec{u}[/latex]
- [latex](c(d\vec{u})) = (cd)(\vec{u})[/latex]
- [latex]1\cdot \vec{u} = \vec{u}[/latex]
Example 1: Determine whether [latex]\vec{y} = \begin{bmatrix}2\\-2\\4\end{bmatrix}[/latex] can be written as a linear combination of [latex]\vec{v_{1}} = \begin{bmatrix}-1\\3\\0\end{bmatrix}[/latex] and [latex]\vec{v_{2}} = \begin{bmatrix}2\\-5\\1\end{bmatrix}[/latex].
Exercise 1: Determine whether [latex]\vec{y} = \begin{bmatrix}1\\3\\-4\end{bmatrix}[/latex] can be written as a linear combination of [latex]\vec{v_{1}} = \begin{bmatrix}1\\2\\-1\end{bmatrix}[/latex] and [latex]\vec{v_{2}} = \begin{bmatrix}3\\-1\\4\end{bmatrix}[/latex].
Definition: A homogeneous linear equation is one whose constant term is equal to zero. A system of linear equations is called homogeneous if each equation in the system is homogeneous. A homogeneous system has the form:
$$\begin{array}{cccc}
a_{11}x_{1}+a_{12}x_{2}+\cdots +a_{1n}x_{n} = 0\\
a_{21}x_{1}+a_{22}x_{2}+\cdots +a_{2n}x_{n} = 0\\
\vdots \\
a_{m1}x_{1}+a_{m2}x_{2}+\cdots +a_{mn}x_{n} = 0
\end{array}$$
Note: $$x_{1}=0, x_{2}=0,\cdots,x_{n} = 0$$ is always a solution to a homogeneous system of equations. We call this the trivial solution.
The zero solution is usually called the trivial solution.
Theorem: If a homogeneous system of linear equations has more variables than equations, then it has a nontrivial solution (in fact, infinitely many).
Theorem: A system of homogeneous equations has a nontrivial solution if and only if the equation has at least one free variable.
Example 2: Determine if the following homogeneous system has a nontrivial solution. Then describe the solution set.
$$\begin{array}{ccc}
x_{1}-3x_{2}+2x_{3} = 0\\
-2x_{1}+x_{2}-3x_{3} = 0\\
5x_{1}+7x_{3} = 0
\end{array}$$
Exercise 2: Determine if the following homogeneous system has a nontrivial solution. Then describe the solution set.
$$\begin{array}{ccc}
-2x_{1}-3x_{2}+2x_{3} = 0\\
-x_{1}+6x_{2}+4x_{3} = 0\\
x_{1}-x_{2}-2x_{3} = 0
\end{array}$$
Definition: 1. The equation of the form [latex]\vec{x} = s\vec{u}[/latex] where [latex]s[/latex] are in [latex]\mathbb{R}[/latex] is called a parametric vector equation of the line. The equation of the form [latex]\vec{x} = s\vec{u} + t\vec{v}[/latex] where [latex]s, t[/latex] are in [latex]\mathbb{R}[/latex] is called a parametric vector equation of the plane when [latex]\vec{u}[/latex] and [latex]\vec{v}[/latex] are not scalar multiple of each others.
2. Whenever a solution set is described explicitly as parametric vector
equations, we say that the solution is in parametric vector form.
Example 3: Suppose the solution set of a certain system of linear equations can be described as [latex]x_{1} = -2 + 3x_{3}, x_{2} = 1 - 2x_{3}[/latex] and [latex]x_{3}[/latex] is free. Use vectors to describe this set as a line in [latex]\mathbb{R}^3[/latex].
Exercise 3: Suppose the solution set of a certain system of linear equations can be described as $$x_{1} = 4 – x_{3}, x_{2} = -1 – 2x_{3}$$ and [latex]x_{3}[/latex] is free. Use vectors to describe this set as a line in [latex]\mathbb{R}^3[/latex].
Note: When a non-homogeneous linear system has many solutions, the general solution can be written in parametric vector form as one vector plus an arbitrary linear combination of vectors that satisfy the corresponding homogeneous system.
Note: Geometrically, we can think of vector addition as a translation. We say that [latex]\vec{v}[/latex] is translated by [latex]\vec{p}[/latex] to [latex]\vec{v} + \vec{p}[/latex]. Moreover, [latex]\vec{p} + t\vec{v}[/latex] is the parametric equation of the line parallel to the vector [latex]\vec{v}[/latex] passing the point corresponding to [latex]\vec{p}[/latex]
Example 4: Describe all solutions of
$$\begin{array}{ccc}
x_{1}-3x_{2}+2x_{3} = 4\\
-2x_{1}+x_{2}-3x_{3} = -3\\
5x_{1}+7x_{3} = 5
\end{array}$$
Exercise 4: Describe all solutions of
$$\begin{array}{cccc}
-2x_{1}-3x_{2}+2x_{3} = -10\\
-x_{1}+6x_{2}+4x_{3} = 1\\
x_{1}-x_{2}-2x_{3} = 3
\end{array}$$
Example 5: Describe all solutions of
$$\begin{array}{ccc}
x_{1}-3x_{2}+2x_{3} = 8\\
x_{1}+2x_{2}-2x_{3} = -3\\
-5x_{1}-5x_{2}+6x_{3} = 4
\end{array}$$
Exercise 5: Describe all solutions of
$$\begin{array}{cccc}
x_{1}-3x_{2}+2x_{3} = -8\\
-x_{1}+8x_{2}+8x_{3} = -7\\
-x_{2}-2x_{3} = 3
\end{array}$$
Group Work 1: Mark each statement True or False. Justify each answer.
a. A homogeneous system is always consistent.
b. A system of homogeneous equations has the trivial solution if and only if
the equation has at least one free variable.
c. The equation [latex]\vec{x} = \vec{p} + t\vec{v}[/latex] describes a line through [latex]\vec{v}[/latex] parallel to [latex]\vec{p}[/latex]
Group Work 2: Consider the following statements about a system of linear
equations with augmented matrix [latex]A[/latex]. In each case either prove the statement or give an example for which it is false.
a. If the system is homogeneous, every solution is trivial.
b. If the system has a nontrivial solution, it cannot be homogeneous.
c. If there exists a trivial solution, the system is homogeneous.
d. If the system is consistent, it must be homogeneous.
Now assume that the system is homogeneous.
e. If there exists a nontrivial solution, there is no trivial solution.
f. If there exists a solution, there are infinitely many solutions.
g. If there exist nontrivial solutions, the row-echelon form of A has a row of zeros.
h. If the row-echelon form of A has a row of zeros, there exist nontrivial
solutions.
i. If a row operation is applied to the system, the new system is also
homogeneous.
Group Work 3: [latex]A[/latex] is the coefficient matrix of a system of equations and
[latex]\vec{b}[/latex] is the constant vector. (a) does the homogeneous system of equations have a
nontrivial solution and (b) does the system of equations have at least one
solution for every possible [latex]\vec{b}[/latex].
(i) [latex]A[/latex] is a [latex]3\times3[/latex] matrix with three pivot positions.
(a)
(b)
(ii) [latex]A[/latex] is a [latex]4\times4[/latex] matrix with three pivot positions.
(a)
(b)
Group Work 4: In each case determine how many solutions (and how many
parameters) are possible for a homogeneous system of four linear equations
in six variables with augmented matrix [latex]A[/latex]. Assume that [latex]A[/latex] has nonzero entries. Give all possibilities.
(a) rank [latex]A=2[/latex]
(b) rank [latex]A=1[/latex]
(c) [latex]A[/latex] has a row of zeros
(d) The echelon form of [latex]A[/latex] has a row of zeros.
Group Work 5: Find all values of [latex]a[/latex] for which the system has nontrivial solutions, and determine all solutions.
$$\begin{array}{cccc}
x_{1}-2x_{2}+x_{3} = 0\\
x_{1}+ax_{2}-3x_{3} = 0\\
-x_{1}+6x_{2}-5x_{3} = 0
\end{array}$$