Exercises – Long Day#
Exercise 1: Properties of Systems of Linear Equations#
We are given the following linear equation system over \(\mathbb R\) in the three unknowns \(x\), \(y\), and \(z\).
Question a#
Is the system homogeneous or inhomogeneous?
Hint
If you are in doubt about what the terms “homogeneous” and “inhomogeneous” mean, then you can find an explanation in the textbook just before Theorem 6.1.1.
Answer
The given linear equation system is inhomogeneous since not all constants on the right-hand side of the equations are equal to zero.
Question b#
Is the triple \((-1,-1,-3) \in \mathbb{R}^3\) a solution to the system? What about the triple \((0,0,1)\)?
Answer
The triple \((-1,-1,-3)\) is not a solution to the system (the first equation is fulfilled but not the second equation).
The triple \((0,0,1)\) is a solution to the system.
Question c#
We still consider the linear equation system given in the beginning of the exercise. What is its corresponding homogeneous linear equation system?
Hint
In Theorem 6.1.2 in the textbook the corresponding homogeneous system is described.
Answer
Question d#
We are given that the triple \((27,1,-74)\) is a solution to the homogeneous system from Question c. Now use the result from Question b to find another solution to the given inhomogeneous linear equation system.
Hint
If needed, first read Theorem 6.1.2 in the textbook.
Answer
The triple \((27,1,-73)\) is a solution to the inhomogeneous linear equation system.
Exercise 2: Augmented Matrix and Row Operations#
We define the following matrix
and the following vector
Question a#
Write out the system of linear equations over \(\mathbb R\) in the unknowns \(x_1\), \(x_2\), and \(x_3\) that has the augmented matrix \([{\mathbf A}|{\mathbf b}]\).
Answer
Question b#
Determine the matrix that you get by performing the row operation \(R_3 \leftarrow R_3+2R_1\) on the matrix \([\mathbf{A}|\mathbf{b}]\).
Hint
If you are in doubt about what the row operation \(R_3 \leftarrow R_3+2R_1\) exactly means, then you can find a decsription just before Example 6.2.4 in the textbook. In Example 6.2.4 a similar row operation is being performed in a specific case.
Answer
Question c#
Now consider the matrix that was found in the previous question (after the row operation) and perform the row operation \(R_2 \leftarrow (1/4)\cdot R_2\) followed by the row operations \(R_1 \leftarrow R_1-3R_2\) and \(R_3 \leftarrow R_3-6R_2\). What is the result?
Answer
Question d#
Determine the reduced row-echelon form of the matrix \([\mathbf{A}|\mathbf{b}]\).
Hint
You do not have to start all over. You can just continue with the matrix that was the answer to Question c.
Answer
After the row operation \(R_3 \leftarrow (1/5)\cdot R_3\) followed by \(R_2 \leftarrow R_2+R_3\) and then \(R_1 \leftarrow R_1-7R_3\) we arrive at the matrix
This matrix is in the wanted reduced row-echelon form.
Question e#
Describe and solve the linear equation system over \(\mathbb R\) that has the reduced row-echelon form found in Question d as its augmented matrix. Check that the solution you find is a solution to the original linear equation system that had the augmented matrix \([{\mathbf A}|{\mathbf b}].\)
Answer
The equation system belonging to the found reduced row-echelon form is
The solution is the triple \((-2/5,1/5,1/5).\) By insertion we see that the triple is a solution to the original system. This is just as it should be since Theorem 6.2.1 in the textbook predicts that the solution set to a linear equation system does not change when elementary row operations are performed.
Exercise 3: Reduced Row-Echelon Forms#
The following matrices are defined
Question a#
Which of these matrices are in row-echelon form? Which are in reduced row-echelon form?
Hint
The definition of a matrix in reduced row-echelon form is given in Definition 6.3.1 in the textbook. Right after this definition it is described what a matrix in row-echelon form fulfills.
Answer
The matrices \({\mathbf A}\) and \({\mathbf D}\) are in row-echelon form. The others are not. Only matrix \({\mathbf D}\) is in reduced row-echelon form.
Question b#
Determine the reduced row-echelon form of each of the given matrices \({\mathbf A}\), \({\mathbf B}\), \({\mathbf C}\), and \({\mathbf D}\).
Answer
The following sequence of row operations leads to the reduced row-echelon forms of the matrices:
Matrix \(\mathbf D\) is already in reduced row-echelon form.
Exercise 4: Reduced Row-Echelon Form of \(4\times 5\) Matrix#
We define the matrix
How many non-zero rows does the reduced row-echelon form of \(\mathbf A\) have?
Answer
The reduced row-echelon form of a given matrix \(\mathbf A\) is
It has 3 non-zero rows. This number is by the way called the rank of the matrix (see Definition 6.3.2).
Exercise 5: Relation between Row-Echelon Form and Reduced Row-Echelon Form#
We define the following matrix in row-echelon form:
Question a#
Without carrying out any computations, justify that \(\mathbf B\) will have a reduced row-echelon form like this:
for certain numbers \(a,b,c \in \mathbb{R}\).
Question b#
Consider why it more generally applies that the number of non-zero rows in a matrix in row-echelon form is equal to the number of non-zero rows in its reduced row-echelon form.
Exercise 6: Examples of Systems of Linear Equations#
Question a#
Provide an example of an inhomogeneous linear equation system over \(\mathbb R\) in the three unknowns \(x_1\), \(x_2\), and \(x_3\) without solutions.
Hint
Corollary 6.4.3 describes when an inhomogeneous linear equation system has no solutions.
Question b#
Can a homogeneous linear equation system have no solutions?
Question c#
Provide an example of a homogeneous linear equation system over \(\mathbb R\) with at least two equations and at least two solutions.
Exercise 7: A System of Four Equations with Four Unknowns#
We again consider the matrix
from Exercise 4. We are informed that the matrix is the augmented matrix of a linear equation system over \(\mathbb R\) in the unknowns \(x_1,x_2,x_3\), and \(x_4\).
Question a#
Determine a solution to the system that fulfills \(x_4=0\). Also determine a solution to the system that fulfills \(x_4=1\).
Question b#
We are given a real number \(t\). Determine a solution to the system that fulfills \(x_4=t\).
Exercise 8: Reduced Row-Echelon Form#
In this exercise, for a given \(m \times (n+1)\) matrix \({\mathbf B}\) we will with the symbol \({\mathbf B}^{(1)}\) (respectively, \({\mathbf B}^{(n+1)}\)) denote the \(m \times n\) matrix that one achieves by deleting the first (respectively, last) column of \(\mathbf B\).
Question a#
Find an example of a \(2 \times 3\) matrix \(\mathbf B\) with coefficients in \(\mathbb C\) such that \(\mathbf B\) is in reduced row-echelon form, while the \(2 \times 2\) matrix \({\mathbf B}^{(1)}\) is not. Is the matrix \({\mathbf B}^{(3)}\) in reduced row-echelon form?
Question b#
About a given \(m \times (n+1)\) matrix \({\mathbf B}\) we are informed that \(\mathbf B\) is in reduced row-echelon form. Justify that the \(m \times n\) matrix \({\mathbf B}^{(n+1)}\) is also in reduced row-echelon form.
Hint
A matrix in reduced row-echelon form fulfills the four requirements given in Definition 6.3.1 in the textbook. Try to realise that if \({\mathbf B}\) fulfills these requirements, then \({\mathbf B}^{(n+1)}\) (where only the last column of \({\mathbf B}\) has been removed) does as well.
Exercise 9: Square Matrix and corresponding System of Linear Equations#
A matrix \(\mathbf A\) is called a square matrix if it contains just as many rows as columns. In other words, the matrix \(\mathbf A\) is square precisely if \({\mathbf A} \in \mathbb{F}^{n \times n}\) for a natural number \(n\).
Question a#
Given a square matrix \({\mathbf A} \in \mathbb{R}^{4 \times 4}\) and a vector \(\mathbf b\in\mathbb R^4\). We are informed that the reduced row-echelon form of \(\mathbf A\) is:
Show that the system of linear equations over \(\mathbb R\) with the augmented matrix \({[{\mathbf A}|{\mathbf b}]} \in \mathbb{R}^{4 \times 5}\) has exactly one solution.
Hint
We are given that the reduced row-echelon form of \({\mathbf A}\) is
Now try to utilize this result to realise that the reduced row-echelon form of the augmented matrix \({[{\mathbf A}|{\mathbf b}]}\) looks as follows:
for some \(c_1,c_2,c_3,c_4 \in \mathbb{R}.\)