Does the equation Ax 0 have a unique solution?

From Theorem 44 we know that Ax = 0 implies that x = 0 necessarily, if and only if all the columns aj of A are linearly independent. That is, x = 0 is the unique solution to Ax = 0 if and only if rank(A) = n.

What does it mean if Ax B has a unique solution?

Given a matrix A and a vector B, a solution of the system AX = B is a vector X which satisfies the equation AX = B. If the null space of A is non-trivial, then the system AX = B has more than one solution. The system AX = B has a unique solution provided dim(N(A)) = 0.

How do you prove a linear system has a unique solution?

In a set of linear simultaneous equations, a unique solution exists if and only if, (a) the number of unknowns and the number of equations are equal, (b) all equations are consistent, and (c) there is no linear dependence between any two or more equations, that is, all equations are independent.

Does equation Ax 0 have a non-trivial solution justify your answer?

Answer: False. If x is not equal to the zero vector, and Ax = 0, then x is a nontrivial solution.

What is the difference between Ax B and ax 0?

Ax = 0 is a homogeneous equations and Ax = b = 0 is a nonhomogeneous equation.

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 Ax  b 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 Ax  b has exactly one solution for every possible b.

How do you know if Ax B has a solution for every 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.

What is the formula for unique solution?

Condition for Unique Solution to Linear Equations A system of linear equations ax + by + c = 0 and dx + ey + g = 0 will have a unique solution if the two lines represented by the equations ax + by + c = 0 and dx + ey + g = 0 intersect at a point. i.e., if the two lines are neither parallel nor coincident.

What is unique solution in linear equation?

The unique solution of a linear equation means that there exists only one point, on substituting which, L.H.S and R.H.S of an equation become equal. The linear equation in one variable has always a unique solution. For example, 3m =6 has a unique solution m = 2 for which L.H.S = R.H.S.

Does the equation Ax B have a solution for all possible b?

The equation Ax = b is solvable for every b. There are n − r = n − m free variables, so there are n − m special solutions to Ax = 0.

Does the equation Ax B 0 have a nontrivial solution and B does the equation Ax d B have at least one solution for every possible b?

The equation Ax  0 does have non-trivial solutions because not every column of A has a pivot position. The equation Ax  0 does have non-trivial solutions because not every column of A has a pivot position.

What happens when Ax 0?

A solution x is non-trivial is x = 0. The homogeneous system Ax = 0 has a non-trivial solution if and only if the equation has at least one free variable (or equivalently, if and only if A has a column with no pivots).

How do you know if Ax has a unique solution?

Then Ax = b has a unique solution if and only if the only solution of Ax = 0 is x = 0. Let A = [A1,A2,,An]. A rephrasing of this is (in the square case) Ax = b has a unique solution exactly when {A1,A2,,An} is a linearly independent set. Click to see full answer.

When does ax = b have a solution?

Ax = b has a solution if and only if b is a linear combination of the columns of A. Note: If A does not have a pivot in every row, that does not mean that Ax = b does not have a solution for some given vector b. It just means that there are some vectors b for which Ax = b does not have a solution.

How do you know if a matrix has a unique solution?

Let A be a square n × n matrix. Then Ax = b has a unique solution if and only if the only solution of Ax = 0 is x = 0. Let A = [A1,A2,,An]. A rephrasing of this is (in the square case) Ax = b has a unique solution exactly when {A1,A2,,An} is a linearly independent set.