Often asked: How can it be shown that the equation axequals=b does not have a solution for some choices of b?

How do you know if Ax B has a solution?

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.

Does the equation Ax B have a solution for each B in R4?

Since the matrix doesn’t have pivots in every row, it follows that the system Ax = b doesn’t have a solution for every bR4.

Does the equation Ax B have at least one solution for every possible b?

only the trivial solution (because every column of A has a pivot position) and the equation Axb does have at least one solution for every possible b (because every row of A has a pivot position). In fact, since every column of A has a pivot position, the equation Axb has exactly one solution for every possible b.

How many solutions does the equation Ax B have explain your answer?

The equation Ax = b has an infinite number of solutions. The equation Ax = 0 has a non-zero solution. Null(A) = {0}. A non–zero linear combination of the columns of A is equal to 0.

What is the solution of Ax B?

One way to find a particular solution to the equation Ax = b is to set all free variables to zero, then solve for the pivot variables. The general solution to Ax = b is given by xcomplete = xp + xn, where xn is a generic vector in the nullspace.

Do columns B span R4?

18 By Theorem 4, the columns of B span R4 if and only if B has a pivot in every row. Therefore, Theorem 4 says that the columns of B do NOT span R4. Further, using Theorem 4, since 4(c) is false, 4(a) is false as well, so Bx = y does not have a solution for each y in R4.

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Is ax b consistent for all B?

The equation Ax = b is consistent if the augmented matrix [A b] has a pivot position in every row. Answer: False. The system is inconsistent if [A b] has a pivot in the last (“b“) column. The system is consistent if the matrix A has a pivot in every row.

What is Ax B when does Ax B has a unique solution?

The system AX = B has a unique solution provided dim(N(A)) = 0. Since, by the rank theorem, rank(A) + dim(N(A)) = n (recall that n is the number of columns of A), the system AX = B has a unique solution if and only if rank(A) = n. A linear system of the form AX = 0 is said to be homogeneous.

What kind of equation is Ax B 0?

The standard or ideal form of a linear equation with one variable is ax + b = , where a and b are constants, x is the variable, and a is not equal to . You can solve the equation for x to get x = − b/a. (See Solving a Linear Equation with One Variable for more information.)

How do you write a parametric solution?

If there are m free variables in the homogeneous equation, the solution set can be expressed as the span of m vectors: x = s1v1 + s2v2 + ··· + smvm. This is called a parametric equation or a parametric vector form of the solution. A common parametric vector form uses the free variables as the parameters s1 through sm.

How do you tell if a system of equations has no solution or infinitely many?

A system of linear equations has one solution when the graphs intersect at a point. No solution. A system of linear equations has no solution when the graphs are parallel. Infinite solutions.

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Is Ax BA a vector equation?

The equation Ax=b is referred to as a vector equation. 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 [ A b ] has a pivot position in every row.

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