1.10.1E: Homogeneous Equations

Exercise \(\PageIndex{1}\)

Consider the following statements about a system of linear equations with augmented matrix \(A\). In each case either prove the statement or give an example for which it is false.

1. If the system is homogeneous, every solution is trivial.
2. If the system has a nontrivial solution, it cannot be homogeneous.
3. If there exists a trivial solution, the system is homogeneous.
4. If the system is consistent, it must be homogeneous.

Now assume that the system is homogeneous.

5. If there exists a nontrivial solution, there is no trivial solution.
6. If there exists a solution, there are infinitely many solutions.
7. If there exist nontrivial solutions, the row-echelon form of \(A\) has a row of zeros.
8. If the row-echelon form of \(A\) has a row of zeros, there exist nontrivial solutions.
9. If a row operation is applied to the system, the new system is also homogeneous.

Answer

1. False. \(A = \left[ \begin{array}{rrr|r} 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 \end{array} \right]\)
2. False. \(A = \left[ \begin{array}{rrr|r} 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 0 \end{array} \right]\)
3. False. \(A = \left[ \begin{array}{rr|r} 1 & 0 & 0 \\ 0 & 1 & 0 \end{array} \right]\)

Exercise \(\PageIndex{2}\)

In each of the following, find all values of \(a\) for which the system has nontrivial solutions, and determine all solutions in each case.

1. \( \begin{array}[t]{rlrlrcr} x & - & 2y & + & z & = & 0 \\ x & + & ay & - & 3z & = & 0 \\ -x & + & 6y & - & 5z & = & 0 \\ \end{array}\)
2. \( \begin{array}[t]{rlrlrcr} x & + & 2y & + & z & = & 0 \\ x & + & 3y & + & 6z & = & 0 \\ 2x & + & 3y & + & az & = & 0 \\ \end{array}\)
3. \( \begin{array}[t]{rlrlrcr} x & + & y & - & z & = & 0 \\ & & ay & - & z & = & 0 \\ x & + & y & + & az & = & 0 \\ \end{array}\)
4. \( \begin{array}[t]{rlrlrcr} ax & + & y & + & z & = & 0 \\ x & + & y & - & z & = & 0 \\ x & + & y & + & az & = & 0 \\ \end{array}\)

Answer

2. \(a = -3\), \(x = 9t\), \(y = -5t\), \(z = t\)
3. \(a = 1\), \(x = -t\), \(y = t\), \(z = 0\); or \(a = -1\), \(x = t\), \(y = 0\), \(z = t\)

Exercise \(\PageIndex{3}\)

Let \(\mathbf{x} = \left[ \begin{array}{r} 2 \\ 1 \\ -1 \end{array} \right]\), \(\mathbf{y} = \left[ \begin{array}{r} 1 \\ 0 \\ 1 \end{array} \right]\), and
\(\mathbf{z} = \left[ \begin{array}{r} 1 \\ 1 \\ -2 \end{array} \right]\). In each case, either write \(\mathbf{v}\) as a linear combination of \(\mathbf{x}\), \(\mathbf{y}\), and \(\mathbf{z}\), or show that it is not such a linear combination.

1. \(\mathbf{v} = \left[ \begin{array}{r} 0 \\ 1 \\ -3 \end{array} \right]\)
2. \(\mathbf{v} = \left[ \begin{array}{r} 4 \\ 3 \\ -4 \end{array} \right]\)
3. \(\mathbf{v} = \left[ \begin{array}{r} 3 \\ 1 \\ 0 \end{array} \right]\)
4. \(\mathbf{v} = \left[ \begin{array}{r} 3 \\ 0 \\ 3 \end{array} \right]\)

Answer

2. Not a linear combination.
3. \(\mathbf{v} = \mathbf{x} + 2\mathbf{y} - \mathbf{z}\)

Exercise \(\PageIndex{4}\)

In each case, either express \(\mathbf{y}\) as a linear combination of \(\mathbf{a}_1\), \(\mathbf{a}_2\), and \(\mathbf{a}_3\), or show that it is not such a linear combination. Here:

\[\mathbf{a}_1 = \left[ \begin{array}{r} -1 \\ 3 \\ 0 \\ 1 \end{array} \right], \ \mathbf{a}_2 = \left[ \begin{array}{r} 3 \\ 1 \\ 2 \\ 0 \end{array} \right], \mbox{ and } \mathbf{a}_3 = \left[ \begin{array}{r} 1 \\ 1 \\ 1 \\ 1 \end{array} \right] \nonumber \]

1. \(\mathbf{y} = \left[ \begin{array}{r} 1 \\ 2 \\ 4 \\ 0 \end{array} \right]\)
2. \(\mathbf{y} = \left[ \begin{array}{r} -1 \\ 9 \\ 2 \\ 6 \end{array} \right]\)

Answer

2. \(\mathbf{y} = 2\mathbf{a}_1 - \mathbf{a}_2 + 4\mathbf{a}_3\).

Exercise \(\PageIndex{5}\)

For each of the following homogeneous systems, find a set of basic solutions and express the general solution as a linear combination of these basic solutions.

1. \( \begin{array}[t]{rlrlrlrlrcr} x_1 & + & 2x_2 & - & x_3 & + & 2x_4 & + & x_5 & = & 0 \\ x_1 & + & 2x_2 & + & 2x_3 & & & + & x_5 & = & 0 \\ 2x_1 & + & 4x_2 & - & 2x_3 & + & 3x_4 & + & x_5 & = & 0 \end{array}\)
2. \( \begin{array}[t]{rlrlrlrlrcr} x_1 & + & 2x_2 & - & x_3 & + & x_4 & + & x_5 & = & 0 \\ -x_1 & - & 2x_2 & + & 2x_3 & & & + & x_5 & = & 0 \\ -x_1 & - & 2x_2 & + & 3x_3 & + & x_4 & + & 3x_5 & = & 0 \end{array}\)
3. \( \begin{array}[t]{rlrlrlrlrcr} x_1 & + & x_2 & - & x_3 & + & 2x_4 & + & x_5 & = & 0 \\ x_1 & + & 2x_2 & - & x_3 & + & x_4 & + & x_5 & = & 0 \\ 2x_1 & + & 3x_2 & - & x_3 & + & 2x_4 & + & x_5 & = & 0 \\ 4x_1 & + & 5x_2 & - & 2x_3 & + & 5x_4 & + & 2x_5 & = & 0 \end{array}\)
4. \( \begin{array}[t]{rlrlrlrlrcr} x_1 & + & x_2 & - & 2x_3 & - & 2x_4 & + & 2x_5 & = & 0 \\ 2x_1 & + & 2x_2 & - & 4x_3 & - & 4x_4 & + & x_5 & = & 0 \\ x_1 & - & x_2 & + & 2x_3 & + & 4x_4 & + & x_5 & = & 0 \\ -2x_1 & - & 4x_2 & + & 8x_3 & + &10x_4 & + & x_5 & = & 0 \end{array}\)

Answer

2. \(r \left[ \begin{array}{r} -2 \\ 1 \\ 0 \\ 0 \\ 0 \end{array} \right] + s \left[ \begin{array}{r} -2 \\ 0 \\ -1 \\ 1 \\ 0 \end{array} \right] + t \left[ \begin{array}{r} -3 \\ 0 \\ -2 \\ 0 \\ 1 \end{array} \right]\)
3. \(s \left[ \begin{array}{r} 0 \\ 2 \\ 1 \\ 0 \\ 0 \end{array} \right] + t \left[ \begin{array}{r} -1 \\ 3 \\ 0 \\ 1 \\ 0 \end{array} \right]\)

Exercise \(\PageIndex{6}\)

1. Does Theorem \(\PageIndex{1}\) imply that the system \(\left \{ \begin{array}{rcl} -z + 3y & = & 0 \\ 2x - 6y & = & 0 \end{array} \right.\) has nontrivial solutions? Explain.
2. Show that the converse to Theorem [thm:001473] is not true. That is, show that the existence of nontrivial solutions does not imply that there are more variables than equations.

Answer

The system in (a) has nontrivial solutions.

Exercise \(\PageIndex{7}\)

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 \(A\). Assume that \(A\) has nonzero entries. Give all possibilities.

Rank \(A = 2\). Rank \(A = 1\). \(A\) has a row of zeros. The row-echelon form of \(A\) has a row of zeros.

Answer

By Theorem \(\PageIndex{2}\), there are \(n - r = 6 - 1 = 5\) parameters and thus infinitely many solutions.

If \(R\) is the row-echelon form of \(A\), then \(R\) has a row of zeros and 4 rows in all. Hence \(R\) has \(r = rank \; A = 1\), \(2\), or \(3\). Thus there are \(n - r = 6 - r = 5\), \(4\), or \(3\) parameters and thus infinitely many solutions.

Exercise \(\PageIndex{8}\)

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Answer

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The graph of an equation \(ax + by + cz = 0\) is a plane through the origin (provided that not all of \(a\), \(b\), and \(c\) are zero). Use Theorem [thm:001473] to show that two planes through the origin have a point in common other than the origin \((0, 0, 0)\).

Exercise \(\PageIndex{1}\)

1. Show that there is a line through any pair of points in the plane. [Hint: Every line has equation \(ax + by + c = 0\), where \(a\), \(b\), and \(c\) are not all zero.]
2. Generalize and show that there is a plane \(ax + by + cz + d = 0\) through any three points in space.

Answer

2. That the graph of \(ax + by + cz = d\) contains three points leads to 3 linear equations homogeneous in variables \(a\), \(b\), \(c\), and \(d\). Apply Theorem [thm:001473].

Exercise \(\PageIndex{10}\)

The graph of

\[a(x^2 + y^2) + bx + cy + d = 0 \nonumber \]

is a circle if \(a \neq 0\). Show that there is a circle through any three points in the plane that are not all on a line.

Exercise \(\PageIndex{11}\)

Consider a homogeneous system of linear equations in \(n\) variables, and suppose that the augmented matrix has rank \(r\). Show that the system has nontrivial solutions if and only if \(n > r\).

Answer

There are \(n - r\) parameters (Theorem \(\PageIndex{2}, so there are nontrivial solutions if and only if \(n - r > 0\).

Exercise \(\PageIndex{12}\)

If a consistent (possibly nonhomogeneous) system of linear equations has more variables than equations, prove that it has more than one solution.