Sample Solution from
Linear Algebra and Its Applications (5th Edition)
5th Edition
ISBN: 9780321982384
Chapter 1
Problem 1SE
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Textbook Problem

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 ℝm.
  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 ℝm, 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 ℝm.
  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 ℝm, 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 ℝm, 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.

Expert Solution

a)

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

Answer:

The given statement is false.

Explanation of Solution

Explanation:

Reason for the statement to be false:

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

b)

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

Answer:

The given statement is false.

Explanation of Solution

Explanation:

Reason for the statement to be false:

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

c)

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

Answer:

The given statement is true.

Explanation of Solution

Explanation:

Reason for the statement to be true:

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

d)

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

Answer:

The given statement is false.

Explanation of Solution

Explanation:

Reason for the statement to be false:

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

x1+x2=1x2=5x1+x2=2

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

e)

To determine

To mark:

The given statement “If an augmented matrix [Ab] is transformed into [Cd] by elementary row operations, then the equations Ax=b and Cx=d have exactly the same solution sets” is true or false.

Answer

Answer:

The given statement is true.

Explanation of Solution

Explanation:

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.
Expert Solution

f)

To determine

To mark:

The given statement “If a system Ax=b has more than one solution, then the system Ax=0 ” is true or false.

Answer

Answer:

The given statement is true.

Explanation of Solution

Explanation:

Reason for the statement to be true:

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

g)

To determine

To mark:

The given statement “If A is an m×n matrix and the equation Ax=b is consistent for some b, then the columns of A span Rm ” is true or false.

Answer

Answer:

The given statement is false.

Explanation of Solution

Explanation:

Reason for the statement to be false:

  • For all values of b, the equation Ax=b is consistent.
Expert Solution

h)

To determine

To mark:

The given statement “If an augmented matrix [Ab] can be transformed by elementary row operations into reduced echelon form, then the equation Ax=b is consistent” is true or false.

Answer

Answer:

The given statement is false.

Explanation of Solution

Explanation:

Reason for the statement to be false:

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

i)

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

Answer:

The given statement is true.

Explanation of Solution

Explanation:

Reason for the statement to be true:

  • The reduced echelon form of the matrices is unique.
Expert Solution

j)

To determine

To mark:

The given statement “The equation Ax=0 has the trivial solution if and only if there are no free variables” is true or false.

Answer

Answer:

The given statement is false.

Explanation of Solution

Explanation:

Reason for the statement to be false:

  • Every equation of Ax=0 has a trivial solution whether there is free variables or not.
Expert Solution

k)

To determine

To mark:

The given statement “If A is an m×n matrix and the equation Ax=b is consistent for every b in Rm , then A has m pivot columns” is true or false.

Answer

Answer:

The given statement is true.

Explanation of Solution

Explanation:

Reason for the statement to be true:

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

l)

To determine

To mark:

The given statement “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 Rm ” is true or false.

Answer

Answer:

The given statement is false.

Explanation of Solution

Explanation:

Reason for the statement to be false:

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

m)

To determine

To mark:

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

Answer

Answer:

The given statement is true.

Explanation of Solution

Explanation:

Reason for the statement to be true:

  • The row-reduced echelon form must be a n×n identity matrix.
Expert Solution

n)

To determine

To mark:

The given statement “If 3×3 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

Answer:

The given statement is true.

Explanation of Solution

Explanation:

Reason for the statement to be true:

  • The matrix A is transformed first into a 3×3 identity matrix, and then it is transformed to B.
Expert Solution

o)

To determine

To mark:

The given statement “If A is an m×n matrix, if the 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” is true or false.

Answer

Answer:

The given statement is true.

Explanation of Solution

Explanation:

Reason for the statement to be true:

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

p)

To determine

To mark:

The given statement “If A and B are row equivalent m×n matrices and if the columns of A span Rm , then so do the columns of B” is true or false.

Answer

Answer:

The given statement is true.

Explanation of Solution

Explanation:

Reason for the statement to be true:

  • Refer theorem 4.
  • For B, all the columns span Rm .
Expert Solution

q)

To determine

To mark:

The given statement “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” is true or false.

Answer

Answer:

The given statement is false.

Explanation of Solution

Explanation:

Reason for the statement to be false:

  • Example: Let the 3 vectors be (1,0,0) , (0,1,0) , and (1,1,0) . Also, the third vector is the total of the first and second vectors.
Expert Solution

r)

To determine

To mark:

The given statement “If {u,v,w} is linearly independent, then u, v, and w are not in R2 ” is true or false.

Answer

Answer:

The given statement is true.

Explanation of Solution

Explanation:

Reason for the statement to be true:

  • Any three vectors that form a set in R2 must be linearly dependent.
Expert Solution

s)

To determine

To mark:

The given statement “In some cases, it is possible for four vectors to span R5 ” is true or false.

Answer

Answer:

The given statement is false.

Explanation of Solution

Explanation:

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 R5 .
Expert Solution

t)

To determine

To mark:

The given statement “If u and v are in Rm , then −u is in Span {u,v} ” is true or false.

Answer

Answer:

The given statement is true.

Explanation of Solution

Explanation:

Reason for the statement to be true:

  • The linear combination of vector u and vector v is vector −u .
Expert Solution

u)

To determine

To mark:

The given statement “If u, v, and w are nonzero vectors in R2 , then w is a linear combination of u and v” is true or false.

Answer

Answer:

The given statement is false.

Explanation of Solution

Explanation:

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.
Expert Solution

v)

To determine

To mark:

The given statement “If w is a linear combination of u and v in Rn , then u is a linear combination of v and w” is true or false.

Answer

Answer:

The given statement is false.

Explanation of Solution

Explanation:

Reason for the statement to be false:

  • Let w=2v , then w=0u+2v . 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.
Expert Solution

w)

To determine

To mark:

The given statement “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” is true or false.

Answer

Answer:

The given statement is false.

Explanation of Solution

Explanation:

Reason for the statement to be false:

  • It is true if the vector v1 is not equal to 0.
Expert Solution

x)

To determine

To mark:

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

Answer

Answer:

The given statement is true.

Explanation of Solution

Explanation:

Reason for the statement to be true:

  • Function is an alternative word for transformation.
Expert Solution

y)

To determine

To mark:

The given statement “If A is a 6×5 matrix, the linear transformation x↦Ax cannot map R5 onto R6 ” is true or false.

Answer

Answer:

The given statement is true.

Explanation of Solution

Explanation:

Reason for the statement to be true:

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

z)

To determine

To mark:

The given statement “If A is an m×n matrix with m pivot columns, then the linear transformation x↦Ax is a one-to-one mapping” is true or false.

Answer

Answer:

The given statement is false.

Explanation of Solution

Explanation:

Reason for the statement to be false:

  • The matrix contains pivots in all columns. So, m<n is not valid.
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