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Does the equation Ax 0 have a nontrivial solution?

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Asked By: Kacie Piepjahn | Last Updated: 7th June, 2020
The homogeneous system Ax 0 always has the trivial solution, x 0. Nonzero vector solutions are called nontrivial solutions.





Just so, what is a nontrivial solution?

A solution or example that is not trivial. Often, solutions or examples involving the number zero are considered trivial. Nonzero solutions or examples are considered nontrivial. For example, the equation x + 5y = 0 has the trivial solution (0, 0). Nontrivial solutions include (5, –1) and (–2, 0.4).

Similarly, what is the solution set of the homogeneous system Ax 0? Thus, the solution set to Ax = 0 is Span{u,v,w}, or parametrically, x = ru + sv + tw where r,s,t ∈ R are parameters. Definition The solution set of a homogeneous equation Ax = 0 is called the kernel of A: ker A := {x ∈ Rn |Ax = 0}.

Consequently, what it means for a system Ax 0 to have infinitely many solutions?

So if det (A) ≠ 0, then AX = B has exactly one solution. If det (A) = 0, then AX = B has infinite solutions or no solutions. The homogeneous system always has the trivial solution of X = 0. AX = 0 has infinitely many solutions.

What is a homogeneous system?

A system of linear equations is homogeneous if all of the constant terms are zero: A homogeneous system is equivalent to a matrix equation of the form. where A is an m × n matrix, x is a column vector with n entries, and 0 is the zero vector with m entries.

What is the trivial solution for a system of homogeneous equations?

Definition TSHSE Trivial Solution to Homogeneous Systems of Equations. Suppose a homogeneous system of linear equations has n variables. The solution x1=0 x 1 = 0 , x2=0 x 2 = 0 , …, xn=0 x n = 0 (i.e. x=0 ) is called the trivial solution.

What is a trivial solution in linear algebra?

Definition. A vector is called trivial if all its coordinates are 0, i. e. if it is the zero vector. In Linear Algebra we are not interested in only finding one solution to a system of linear equations. In particular, homogeneous systems of equations (see above) are very important.

What is a null solution?

The null solution (or as it’s more commonly called, the complementary solution) is the solution to the homogeneous equation. In this case, it is y = Ce4t. The particular solution is a solution to the nonhomogeneous equation.

What is a unique solution in linear equations?

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 do you mean by trivial solution?

Trivial. A solution or example that is ridiculously simple and of little interest. Often, solutions or examples involving the number 0 are considered trivial. Nonzero solutions or examples are considered nontrivial. For example, the equation x + 5y = 0 has the trivial solution x = 0, y = 0.

What is the condition for non zero solution?

A nxn nonhomogeneous system of linear equations has a unique nontrivial solution if and only if its determinant is nonzero. If this determinant is zero, then the system has either no nontrivial solutions or an infinite number of solutions.

Can the solution set of Ax B be a plane through the origin?

ALWAYS consistent because you can always write down the solution; talking all variable to be a zero vector. If b cannot equal zero, can Ax=b be a plane through the origin? No. The equation of a plane through the origin has Ax=b and MUST = 0.

What is the rank of a matrix?

The rank of a matrix is defined as (a) the maximum number of linearly independent column vectors in the matrix or (b) the maximum number of linearly independent row vectors in the matrix. Both definitions are equivalent. For an r x c matrix, If r is less than c, then the maximum rank of the matrix is r.

What is a consistent system?

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 . When you graph the equations, both equations represent the same line.

What is linear homogeneous equation?

A homogeneous linear differential equation is a differential equation in which every term is of the form y ( n ) p ( x ) y^{(n)}p(x) y(n)p(x) i.e. a derivative of y times a function of x. In fact, looking at the roots of this associated polynomial gives solutions to the differential equation.

What is a free variable in linear algebra?

Free and Basic Variables. A variable is a basic variable if it corresponds to a pivot column. Otherwise, the variable is known as a free variable. In order to determine which variables are basic and which are free, it is necessary to row reduce the augmented matrix to echelon form.

Why does the equation Ax B have a solution?

The first thing to know is what Ax means: it means we are multiplying the matrix A times the vector x. Solving Ax = b is the same as solving the system described by the augmented matrix [A|b]. • Ax = b has a solution if and only if b is a linear combination of the columns of A.

Is every homogeneous linear system consistent?

A homogeneous system must have at least one solution, . That is, every homogeneous system of linear equations is consistent. Note: The solution is called the trivial solution to the homogeneous system.

Is a system consistent if it has a free variable?

(2) A consistent system of linear equations must have a free variable. false Reason: If there are no free variables then the system can still be consistent; it will have a unique solution. Thus one column remains that always contributes a free variable and infinitely many solutions.

What is Ax B when does Ax B 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.

What is a in Ax B?

Definition. If A is an m n matrix, with columns a1,a2,,an, and if x is in Rn, then the product of A and x, denoted by Ax, is the linear combination of the columns of A using the corresponding. entries in x as weights.

What kind of equation is Ax B 0?

Linear equations

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