Use the definition of Ax to write the matrix equation as a vector equation. [-1 6 -6 6 -9 7 -8 7][-5 -2] = [-7 18 31 26]
Write the system first as a vector equation and then as a matrix equation. 8x1 + x2 - 3x3 = 8 5x2 + 4x3 = 0 (a) Write the system as a vector equation where the first equation of the system corresponds to the first row. Select the correct choice below and fill in any answer boxes to complete your choice. (b) Write the system as a matrix equation where the first equation of the system corresponds to the first row. Select the correct choice below and fill in any answer boxes to complete your choice.
Given A and b to the right, write the augmented matrix for the linear system that corresponds to the matrix equation Ax = b. Then solve the system and write the solution as a vector. A = [1 3 -2 -2 -4 2 5 2 6], b = [4 -2 -13] Write the augmented matrix for the linear system that corresponds to the matrix equation Ax=b. Solve the system and write the solution as vector.
Let u = [-10 -8 -7], A = [6 7 16 0 2 -4 1 2 1]. Is u in the subset of R^3 spanned by the columns of A? Why or why not? Select the correct choice below and fill in the answer box to complete your choice. (Type an integer or decimal for each matrix element.) A. No, the reduced row echelon form of the augmented matrix is which is an inconsistent system. B. Yes, multiplying A by the vector writes u as a linear combination of the columns of A.
Let A = [1 -4 -3 -4 4 0 2 4 6] and b = [b1 b2 b3] Show that the equation Ax = b does not have a solution for all possible b, and describe the set of all b for which Ax = b does have a solution. How can it be shown that the equation Ax = b does not have a solution for all possible b? Choose the correct answer below. A. Find a vector b for which the solution to Ax = b is the zero vector. B. Row reduce the matrix A to demonstrate that A does not have a pivot position in every row. C. Find a vector x for which Ax = b is the zero vector. D. Row reduce the matrix A to demonstrate that A has a pivot position in every row. E. Row reduce the augmented matrix [ A b ] to demonstrate that [ A b ] has a pivot position in every row. Describe the set of all b for which Ax = b does have a solution. Type an expression using b1, b2, and b3 as the variables and 1 as the coefficient of b)
Let v1 = [1 0 0 -1], v2 = [0 -1 0 1], v3 = [0 0 1 -1]. Does {v1, v2, v3} span R^4? Why or why not? Choose the correct answer below. A. Yes. When the given vectors are written as the columns of a matrix A, A has a pivot position in every row. B. Yes. Any vector in R4 except the zero vector can be written as a linear combination of these three vectors. C. No. When the given vectors are written as the columns of a matrix A, A has a pivot position in only three rows. D. No. The set of given vectors spans a plane in R^4. Any of the three vectors can be written as a linear combination of t
Let v1 = [0 0 -5], v2 = [0 -4 10], v3 = [6 -3 15]. Does {v1, v2, v3 } span R^3? Why or why not? Choose the correct answer below. A. Yes. Any vector in R^3 except the zero vector can be written as a linear combination of these three vectors. B. No. The set of given vectors spans a plane in R^3. Any of the three vectors can be written as a linear combination of t C. Yes. When the given vectors are written as the columns of a matrix A, A has a pivot position in every row. D. No. When the given vectors are written as the columns of a matrix A, A has a pivot position in only two rows.
A vector b is a linear combination of the columns of a matrix A if and only if the equation Ax = b has at least one solution. Choose the correct answer below. A. True. The equation Ax = b has the same solution set as the equation x1a1 + x2a2 +...+xnan = b B. True. The equation Ax = b is unrelated to whether the vector b is a linear combination of the columns of a matrix A. C. False. If the matrix A is the identity matrix, then the equation Ax = b has at least one solution, but b is not a linear combination of the columns of A. D. False. If the equation Ax = b has infinitely many solutions, then the vector b cannot be a linear combination of the columns of A.
The equation Ax = b is consistent if the augmented matrix [ A b ] has a pivot position in every row. Choose the correct answer below. A. False. If the augmented matrix [A b] has a pivot position in every row, the equation Ax = b may or may not be consistent. One pivot position may be in the column representing b. B. True. If the augmented matrix [A b] has a pivot position in every row, then the equation Ax = b has a solution for each b in Rm. C. True. The pivot positions in the augmented matrix [ A b ] always occur in the columns that represent A. D. False. The augmented matrix [ A b ] cannot have a pivot position in every row because it has more columns than rows.
If the columns of an m×n matrix A span R^m, then the equation Ax = b is consistent for each b in R^m. Choose the correct answer below. A. False. Since the columns of A span R^m, the matrix A has a pivot position in exactly m- 1 rows. B. True. Since the columns of A span R^m, the augmented matrix [A b] has a pivot position in every row. C. True. If the columns of A span R^m, then the equation Ax = b has a solution for each b∈ R^m D. False. If the columns of A span R^m, then the equation Ax = b is inconsistent for each b∈ R^m
Could a set of three vectors in R^4 span all of R^4? Explain. What about n vectors in R^4 when n is less than m? Could a set of three vectors in R^4 span all of R^4? Explain. Choose the correct answer below. A. Yes. Any number of vectors in R^4 will span all of R^4. B. Yes. A set of n vectors in R^m can span R^m when n < m. There is a sufficient number of rows in the matrix A formed by the vectors to have enough pivot points to show that the vectors span R^m C. No. The matrix A whose columns are the three vectors has four rows. To have a pivot in each row, A would have to have at least four columns (one for each pivot). D. No. There is no way for any number of vectors in R^4 to span all of R^4. Could a set of n vectors in R^m span all of R^m when n is less than m? Explain. Choose the correct answer below. A. No. The matrix A whose columns are the n vectors has m rows. To have a pivot in each row, A would have to have at least m columns (one for each pivot). B. Yes. Any number of vectors in R^m will span all of m C. Yes. A set of n vectors in R^m can span R^m if n < m. There is a sufficient number of rows in the matrix A formed by the vectors to have enough pivot points to show that the vectors span R^m D. No. Without knowing values of n and m, there is no way to determine if n vectors in R^m will span all of R^m
Determine if the system has a nontrivial solution. Try to use as few row operations as possible. 4x1 - 6x2 + 13x3 = 0 -4x1 - 10x2 - x3 = 0 8x1 + 4x2 + 14x3 = 0 Choose the correct answer below. A. The system has a nontrivial solution. B. It is impossible to determine. C. The system has only a trivial solution.
Determine if the system has a nontrivial solution. Try to use as few row operations as possible. x1 - 3x2 + 7x2 = 0 -2x1 + 2x2 - 5x3 = 0 x1 + x2 + 9x3 = 0 Choose the correct answer below. A. The system has a nontrivial solution. B. It is impossible to determine. C. The system has only a trivial solution.
Write the solution set of the given homogeneous system in parametric vector form. 3x1 + 3x2 + 6x3 = 0 -9x1 - 9x2 - 18x3 = 0 -3x2 + 6x3 = 0 where the solution set is x = [x1 x2 x3]
Describe all the solutions of Ax = 0 in the parametric vector form, where A is row equivalent to the given matrix [1 2 - 4 6 0 1 - 4 7]
Describe all the solutions of Ax = 0 in parametric vector form, where A is row equivalent to the given matrix [1 4 0 - 5 4 16 0 - 20]
Describe the solutions of the first system of equations below in parametric vector form. Provide a geometric comparison with the solution set of the second system of equations below. 2x1 + 2x2 + 4x3 = 8 -6x1 - 6x2 - 12x3 = -24 -5x2 + 10x3 = 10 2x1 + 2x2 + 4x3 = 0 -6x1 - 6x2 - 12x3 = 0 -5x2 + 10x3 = 0 Describe the solution set, x = [x1 x2 x3] of the first system of equations in parametric vector form. Which option best compares the two systems? A. The solution set of the first system is a line parallel to the line that is the solution set of the second system. B. The solution set of the first system is a line perpendicular to the line that is the solution set of the second system. C. The solution set of the first system is a plane parallel to the plane that is the solution set of the second system. D. The solution set of the first system is a plane parallel to the line that is the solution set of the second system