The equation Ax = b is consistent if the augmented matrix [A b] has a pivot position in every row. Answer: False. The system is consistent if the matrix A has a pivot in every row. How do you know if Ax B has a solution for every B?
Is the equation ax = b consistent for all possible?
Determine if the equation Ax = b is consistent for all possible . If the equation is not consistent for all possible , give a description of the set of all b for which the equation is consistent. I know that the solution is that the equation is consistent for all satisfying via an answer key but I don't understand how to get there.
Can ax = b be consistent for all B in R^3?
Explain why the equation Ax = b cannot be consistent for all b in set of real numbers R^3. Generalize your argument to the case of an arbitrary A with more rows than columns.
What can you say about the ax = b?
When solving Ax = b, that row will represent an equation with a zero on the left side and a possibly nonzero entry of b on the right side. Suppose A is a 4 x 3 matrix and b is a vector in set of real numbers R^4 with the property that Ax = b has a unique solution. What can you say about the reduced echelon form of A? Justify your answer.
Which equation has the same solution set as Ax = b?
The equation Ax = b has the same solution set as the equation x 1 a 1 + x 2 a 2 +x n *a n = b. The equation Ax = b is consistent if the augmented matrix left bracket [A b] has a pivot position in every row.
Why is the equation Ax B not consistent for all B in?
The equation Ax=b may or may not be consistent if the augmented matrix [A b] has a pivot position in every row. If one of the pivot positions occurs in the column that represents b, then the equation is not consistent. If all of the pivot positions occur in columns that represent A, then the equation is consistent.
What does it mean when Ax B is inconsistent?
(1) Ax = b is inconsistent iff rank(A ) = rank[A |b ] iff [A |b ] contains a row in which the only nonzero entry lies in the last column, the b column. (2) Ax = b is consistent iff [A |b ] contains no row in which the only nonzero entry lies in the last column.
Is the equation Ax b consistent for all possible b1 b2 b3?
Therefore, the equation Ax = b is consistent if and only if b1 + 2b2 + b3 = 0, i.e, if the rightmost column is not a pivot column. Hence, the system is not consistent for every possible choice of b; it is inconsistent if b1 + 2b2 + b3 = 0.
How do you determine if a system is consistent or not?
Systems of equations can be classified by the number of solutions. If a system has at least one solution, it is said to be consistent . If a consistent system has exactly one solution, it is independent . If a consistent system has an infinite number of solutions, it is dependent .
How do I know when Ax B is sometimes consistent?
If the columns of an mxn matrix A span R^m, then the equation Ax = b is consistent for each b in R^m.
How do you know if a matrix is inconsistent?
Inconsistent. If a system of equations has no solutions, then it is inconsistent. If the last column (in an augmented matrix) is a pivot column, that is, it has a pivot, then it's inconsistent.
Does the equation Ax B have a solution for all possible b?
Ax = b has a solution if and only if b is a linear combination of the columns of A. Theorem 4 is very important, it tells us that the following statements are either all true or all false, for any m × n matrix A: (a) For every b, the equation Ax = b has a solution.
Under what conditions on b1 b2 does Ax B have a solution?
Explanation: A matrix equation Ax = b has a solution when the last column of the associated aug- mented matrix [A b] is not a pivot column. 0 0 b2 + 2b1 ]. Thus the equation has a solution when b2 + 2b1 = 0 , i.e., (b1, b2) satisfies the linear equation y + 2x = 0 .
For what value of b3 the system of equations is consistent?
So I can say the system is consistent if and only if b3=−2b1+b2.
What is an example of an inconsistent equation?
Inconsistent equations is defined as two or more equations that are impossible to solve based on using one set of values for the variables. An example of a set of inconsistent equations is x+2=4 and x+2=6.
What is meant by consistent equations?
In mathematics and particularly in algebra, a linear or nonlinear system of equations is called consistent if there is at least one set of values for the unknowns that satisfies each equation in the system—that is, when substituted into each of the equations, they make each equation hold true as an identity.
What is meant by consistent equation give example?
A consistent meaning in maths is an equation that has at least one solution in common. Let's take an example of consistent equations as x + y = 6 and x – y = 2 there is one solution in common.
What is trivial and non-trivial?
The word trivial is used for simple and evident concepts or things, such as – topological spaces and groups that have a simple arrangement. The antonym of trivial is non-trivial. It is used to indicate non-obvious statements and easy-to-prove theorems in Mathematics as well as in Engineering.
Do the equation Ax B have at least one solution for every possible b?
Ax = 0 has a nontrivial solution. Ax = b has at least one solution for every possible b.
What is the difference between homogeneous and non-homogeneous system of linear equations?
For a homogeneous system of linear equations either (1) the system has only one solution, the trivial one; (2) the system has more than one solution. For a non-homogeneous system either (1) the system has a single (unique) solution; (2) the system has more than one solution; (3) the system has no solution at all.
Does ax 0 have a nontrivial solution?
Fact. The homogeneous equation Ax = 0 has a nontrivial solution if and only if the equation has at least one free variable.
Which system of equations is consistent if detA 0.?
The system of equations Ax = B is consistent if detA ≠ 0. Here we have:
Is the augmented matrix consistent?
Both the augmented matrix (A ∣ b) and the coefficient matrix A have a rank of 3 - so the system is consistent.
How many variables does a x = b equation have?
I view this as a system of equations. A x = b gives you 3 linear equations with 2 variables.
Is the equation A x = B consistent?
HINT: The equation A x = b is consistent if and only if b is a linear combination of the columns of A, i.e., if and only if b is in the column space of A. What is the maximum possible dimension of the column space of A? What is the dimension of R 3? (If you get the point of this hint, the generalization should be pretty obvious.)
Why is the system consistent?
The system is consistent because the rightmost column of the augmented matrix is not a pivot column.
When u and v are nonzero vectors, Span u v contains only?
When u and v are nonzero vectors, Span { u,v} contain s only the line through u and the line through v and the origin.
What is an example of a linear combination of vectors v1 and v2?
An example of a linear combination of vectors v1 and v2 is the vector (1/2) v1. Choose the correct answer below.
When does an augmented matrix have a solution?
True. An augmented matrix has a solution when the last column can be written as a linear combination of the other columns. A linear system augmented has a solution when the last column of its augmented matrix can be written as a linear combination of the other columns.
What is consistent system?
True, a consistent system is defined as a system that has at least one solution. In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations. The statement is false. Each matrix is row equivalent to one and only one reduced echelon matrix.
How many columns does a 5 times 6 matrix have?
False, because a 5 times 6 matrix has five rows and six columns.
Which algorithm applies only to augmented matrices for a linear system?
The row reduction algorithm applies only to augmented matrices for a linear system.
Who created the solution set of Ax=B?
Exploring the solution set of Ax=b (non homogeneous equations). Created by Sal Khan.
Is xp + N a set?
Thanks for the responses. samkoff has the right idea. xp + N (A) doesn't mean a particular solution, it is still a set, I just unrigorously used the plus operator. A good way to demonstrate union is incorrect is as follows. Assume there is a UNIQUE NON-ZERO SOLUTION. If Union is used then the solution set would be {0,Xp} (since N (A) = {0}) which is obviously not a unique solution.
Does the parallel line to the null space represent the solution vectors in standard form?
Never mind, I realized that the parallel line to the null space doesn't represent the solution vectors in standard form, but rather the solution set as a whole.
