Chapter 1, Problem 1SE

Mark each statement True or False. Justify each answer. (If true, cite appropriate facts or theorems). If false, explain why or give a counterexample that shows why the statement is not true in every case.

1. a. Even’ matrix is row equivalent to a unique matrix in echelon form.
2. b. Any system of n linear equations in n variables has at most n solutions.
3. c. If a system of linear equations has two different solutions, it must have infinitely many solutions.
4. d. If a system of linear equations has no free variables, then it has a unique solution.
5. e. If an augmented matrix [A b] is transformed into [C d] by elementary row operations, then the equations Ax = b and Cx = d have exactly the same solution sets.
6. f. If a system Ax = b has more than one solution, then so does the system Ax = 0.
7. g. If A is an m × n matrix and the equation Ax = b is consistent for some b, then the columns of A span ℝ^m.
8. h. If an augmented matrix [A b] can be transformed by elementary row operations into reduced echelon form, then the equation Ax = b is consistent.
9. i. If matrices A and B are row equivalent, they have the same reduced echelon form.
10. j. The equation Ax = 0 has the trivial solution if and only if there are no free variables.
11. k. If A is an m × n matrix and the equation Ax = b is consistent for every b in ℝ^m, then A has m pivot columns.
12. l. If an m × n matrix A has a pivot position in every row, then the equation Ax = b has a unique solution for each b in ℝ^m.
13. m. If an n × n matrix A has n pivot positions, then the reduced echelon form of A is the n × n identity matrix.
14. n. If 3 x 3 matrices A and B each have three pivot positions, then A can be transformed into B by elementary row operations.
15. ○.       If A is an m × n matrix, if die equation Ax = b has at least two different solutions, and if the equation Ax = c is consistent, then the equation Ax = c has many solutions.
16. p. If A and B are row equivalent m x n matrices and if the columns of A span ℝ^m, then so do the columns of B.
17. q. If none of the vectors in the set S = {v₁. v₂, v₃} in R³ is a multiple of one of the other vectors, then S is linearly independent.
18. r. If {u, v, w} is linearly independent, then u, v, and w are not in ℝ².
19. s. In some cases, it is possible for four vectors to span ℝ⁵
20. t. If u and v are in ℝ^m, then −u is in Span{u, v}.
21. u. If u, v, and w are nonzero vectors in ℝ², then w is a linear combination of u and v.
22. v. If w is a linear combination of u and v in ℝⁿ, then u is a linear combination of v and w.
23. w. Suppose that v₁, v₂, and v₃ are in ℝ⁵, v₂ is not a multiple of v₁, and v₃ is not a linear combination of v₁ and v₂, Then { v₁, v₂, v₃} is linearly independent.
24. x. A linear transformation is a function.
25. y. If A is a 6 × 5 matrix, the linear transformation x ↦ Ax cannot map ℝ⁵ onto ℝ⁶.
26. z. If A is an m × n matrix with m pivot columns, then the linear transformation x ↦ Ax is a one-to-one mapping.

To determine

To mark:

The given statement “Every matrix is a row equivalent to a unique matrix in echelon form” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Reason for the statement to be false:

• A matrix is equal to unique matrix only if it is in reduced echelon form.

To determine

To mark:

The given statement “Any system of n linear equations in n variables has at most n solutions” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Reason for the statement to be false:

• The n linear equations with n variables resulted in many solutions.

To determine

To mark:

The given statement “If a system of linear equations has two different solutions, it must have infinitely many solutions” is true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Reason for the statement to be true:

• Refer the “Existence and Uniqueness theorem”.
• The system contains infinite solutions.

To determine

To mark:

The given statement “If a system of linear equations has two different solutions, it must have infinitely many solutions” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Reason for the statement to be false:

• An example of a system that contains no free variables has no solution is shown below:

$\begin{array}{l}{x}_1+x_2=1\\ x_2=5\\ x_1+x_2=2\end{array}$

• Solution does not exist for the system with the absence of free variables.

To determine

To mark:

The given statement “If an augmented matrix $\left[\begin{array}{cc}A& \text{b}\end{array}\right]$ is transformed into $\left[\begin{array}{cc}C& \text{d}\end{array}\right]$ by elementary row operations, then the equations $A\text{x}=\text{b}$ and $C\text{x}=\text{d}$ have exactly the same solution sets” is true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Reason for the statement to be true:

• Refer the box below the title “definition of elementary row operations”.
• Transformation that resulted in the two matrixes is to be row equivalent.

To determine

To mark:

The given statement “If a system $A\text{x}=\text{b}$ has more than one solution, then the system $A\text{x}=\text{0}$” is true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Reason for the statement to be true:

• Refer theorem 6.
• The equations result in equal number of solutions.

To determine

To mark:

The given statement “If A is an $m\times n$ matrix and the equation $A\text{x}=\text{b}$ is consistent for some b, then the columns of A span ${\text{R}}^m$” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Reason for the statement to be false:

• For all values of b, the equation $A\text{x}=\text{b}$ is consistent.

To determine

To mark:

The given statement “If an augmented matrix $\left[\begin{array}{cc}A& \text{b}\end{array}\right]$ can be transformed by elementary row operations into reduced echelon form, then the equation $A\text{x}=\text{b}$ is consistent” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Reason for the statement to be false:

• Any matrix can be modified into row-reduced echelon form, but not all the matrices are consistent.

To determine

To mark:

The given statement “If matrices A and B are row equivalent, they have the same reduced echelon form” is true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Reason for the statement to be true:

• The reduced echelon form of the matrices is unique.

To determine

To mark:

The given statement “The equation $A\text{x}=\text{0}$ has the trivial solution if and only if there are no free variables” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Reason for the statement to be false:

• Every equation of $A\text{x}=0$ has a trivial solution whether there is free variables or not.

To determine

To mark:

The given statement “If A is an $m\times n$ matrix and the equation $A\text{x}=\text{b}$ is consistent for every b in ${\text{R}}^m$, then A has m pivot columns” is true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Reason for the statement to be true:

• Refer theorem 4.
• Each column of a matrix has one pivot point.

To determine

To mark:

The given statement “If an $m\times n$ matrix A has a pivot position in every row, then the equation $A\text{x}=\text{b}$ has a unique solution for each b in ${\text{R}}^m$” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Reason for the statement to be false:

• The equation contains the minimum number of free variable as 1. So, it has infinite solutions.

To determine

To mark:

The given statement “If an $n\times n$ matrix A has n pivot positions, then the reduced echelon form of A is the $n\times n$ matrix identity matrix” is true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Reason for the statement to be true:

• The row-reduced echelon form must be a $n\times n$ identity matrix.

To determine

To mark:

The given statement “If $3\times 3$ matrices A and B each have three pivot positions, then A can be transformed into B by elementary row operations” is true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Reason for the statement to be true:

• The matrix A is transformed first into a $3\times 3$ identity matrix, and then it is transformed to B.

To determine

To mark:

The given statement “If A is an $m\times n$ matrix, if the equation $A\text{x}=\text{b}$ has at least two different solutions, and if the equation $A\text{x}=\text{c}$ is consistent, then the equation $A\text{x}=\text{c}$ has many solutions” is true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Reason for the statement to be true:

• Refer theorem 6.
• Both equations result in equal number of solutions.

To determine

To mark:

The given statement “If A and B are row equivalent $m\times n$ matrices and if the columns of A span ${\text{R}}^m$, then so do the columns of B” is true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Reason for the statement to be true:

• Refer theorem 4.
• For B, all the columns span ${\text{R}}^m$.

To determine

To mark:

The given statement “If none of the vectors in the set $S=\left\{{\text{v}}_1,{\text{v}}_2,{\text{v}}_3\right\}$ in ${\text{R}}^3$ is a multiple of one of the other vectors, then S is linearly independent” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Reason for the statement to be false:

• Example: Let the 3 vectors be $\left(1,0,0\right)$, $\left(0,1,0\right)$, and $\left(1,1,0\right)$. Also, the third vector is the total of the first and second vectors.

To determine

To mark:

The given statement “If $\left\{\text{u},\text{v},\text{w}\right\}$ is linearly independent, then u, v, and w are not in ${\text{R}}^2$” is true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Reason for the statement to be true:

• Any three vectors that form a set in ${\text{R}}^2$ must be linearly dependent.

To determine

To mark:

The given statement “In some cases, it is possible for four vectors to span ${\text{R}}^5$” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Reason for the statement to be false:

• The four vectors have 4 columns, and it contains the maximum number of pivots as 4 so it cannot span ${\text{R}}^5$.

To determine

To mark:

The given statement “If u and v are in ${\text{R}}^m$, then $-\text{u}$ is in Span $\left\{\text{u},\text{v}\right\}$” is true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Reason for the statement to be true:

• The linear combination of vector u and vector v is vector $-\text{u}$.

To determine

To mark:

The given statement “If u, v, and w are nonzero vectors in ${\text{R}}^2$, then w is a linear combination of u and v” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Reason for the statement to be false:

• If u and v are multiples, then Span {u, v} is a line, and w need not be on that line.

To determine

To mark:

The given statement “If w is a linear combination of u and v in ${\text{R}}^n$, then u is a linear combination of v and w” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Reason for the statement to be false:

• Let $\text{w}=2\text{v}$, then $\text{w}=0\text{u}+2\text{v}$. Here, linear combination of vector u and vector v is vector w, but linear combination of vector v and vector w cannot be vector u.

To determine

To mark:

The given statement “Suppose that ${\text{v}}_1$, ${\text{v}}_2$, and ${\text{v}}_3$ are in ${\text{R}}^5$, ${\text{v}}_2$ is not a multiple of ${\text{v}}_1$, and ${\text{v}}_3$ is not a linear combination of ${\text{v}}_1$ and ${\text{v}}_2$. Then, $\left\{{\text{v}}_1,{\text{v}}_2,{\text{v}}_3\right\}$ is linearly independent” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Reason for the statement to be false:

• It is true if the vector ${\text{v}}_1$ is not equal to 0.

To determine

To mark:

The given statement “A linear transformation is a function” is true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Reason for the statement to be true:

• Function is an alternative word for transformation.

To determine

To mark:

The given statement “If A is a $6\times 5$ matrix, the linear transformation $\text{x}\mapsto A\text{x}$ cannot map ${\text{R}}^5$ onto ${\text{R}}^6$” is true or false.

Answer to Problem 1SE

The given statement is true.

Explanation of Solution

Reason for the statement to be true:

• The matrix with 6 pivot columns is not possible because the matrix has only 5 columns.

To determine

To mark:

The given statement “If A is an $m\times n$ matrix with m pivot columns, then the linear transformation $\text{x}\mapsto A\text{x}$ is a one-to-one mapping” is true or false.

Answer to Problem 1SE

The given statement is false.

Explanation of Solution

Reason for the statement to be false:

• The matrix contains pivots in all columns. So, $\text{m}<\text{n}$ is not valid.