What does it mean if Ax 0?

What does it mean if Ax 0?

A homogeneous system is one that can be written in the form Ax = 0. Equivalently, a homogeneous system is any system Ax = b where x = 0 is a solution (notice that this means that b = 0, so both definitions match). The solution x = 0 is called the trivial solution. A solution x is non-trivial is x = 0.

What is the solution set of Ax 0?

The homogeneous system Ax = 0 always has the trivial solution, x = 0. Nontrivial Solution Nonzero vector solutions are called nontrivial solutions. Consistent system with a free variable has infinitely many solutions.

Does 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 is Ax B matrix?

Note that Ax is defined only if the number of columns of A equals the number of entries in x. The equation Ax = b is called a matrix equation. The equation Ax = b has a solution if and only if b is a linear combination of the columns of A. Theorem 4. Let A be an m×n matrix.

What does ax 0 Matrix mean?

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

Is Ax 0 a vector space?

The solution set of the linear system AX = 0 is a vector space. or, more concisely, AX = 0. If r = n the solution consists of only the single solution X = 0, which is called the trivial solution.

What is the meaning of trivially?

1a : of little worth or importance a trivial objection trivial problems. b : relating to or being the mathematically simplest case specifically : characterized by having all variables equal to zero a trivial solution to a linear equation. 2 : commonplace, ordinary.

What is a trivial statement?

Trivial statements are ones that add very little new content about the topic. For example: I think everyone should learn more about the Egyptian culture.

What is the difference between Ax B and ax 0?

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

Under what conditions the equation Ax B will have 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.

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