Can an overdetermined system have infinite solutions?

Asked by: Mrs. Ruth Goldner I
Score: 4.4/5 (48 votes)

In mathematics, a system of equations is considered overdetermined if there are more equations than unknowns. An overdetermined system is almost always inconsistent (it has no solution) when constructed with random coefficients. ... Such systems usually have an infinite number of solutions.

How many solutions does an undetermined system have?

Theorem 2 (Missing Variable)

A system of m×n linear homogeneous equations with one unknown missing has at least one free variable, hence an infinite number of solutions. Therefore, such a system always has the zero solution and also a nonzero solution.

Do all homogeneous systems have infinite solutions?

Every homogeneous system has either exactly one solution or infinitely many solutions. If a homogeneous system has more unknowns than equations, then it has infinitely many solutions.

How do you know if a system has infinite solutions?

If the two lines have the same y-intercept and the slope, they are actually in the same exact line. In other words, when the two lines are the same line, then the system should have infinite solutions. It means that if the system of equations has an infinite number of solution, then the system is said to be consistent.

26 related questions found

Is 0 0 infinite or no solution?

For an answer to have an infinite solution, the two equations when you solve will equal 0=0 . Here is a problem that has an infinite number of solutions. If you solve this your answer would be 0=0 this means the problem has an infinite number of solutions.

What is an equation with infinite solutions?

No solution would mean that there is no answer to the equation. It is impossible for the equation to be true no matter what value we assign to the variable. Infinite solutions would mean that any value for the variable would make the equation true.

How do you know if a system is homogeneous?

This representation can also be done for any number of equations with any number of unknowns. In general, the equation AX=B representing a system of equations is called homogeneous if B is the nx1 (column) vector of zeros. Otherwise, the equation is called nonhomogeneous.

How do you know if a 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.

Can a homogeneous system have no solutions?

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.

Is the system underdetermined or overdetermined?

The overdetermined case occurs when the system has been overconstrained — that is, when the equations outnumber the unknowns. In contrast, the underdetermined case occurs when the system has been underconstrained — that is, when the number of equations is fewer than the number of unknowns.

Can there ever be multiple least squares solutions?

Yes, linear regression problem can have degenerated solution, i.e. multiple solutions equally good in a sense of the lowest sum of squared residuals. A simple example is to have two identical variables in the equation, such as a temperature in Fahrenheit and Celsius.

What is minimum norm solution?

A vector x∗ satisfying Ax∗ = b is the minimum-norm solution to the system of equations Ax = b if and only if x∗ · y = 0 for all solutions y of the homogeneous system Ay = 0. There's another way to phrase this condition.

What is the formula for no solution?

If (a1/a2) = (b1/b2) ≠ (c1/c2), then there will be no solution. This type of equation is called an inconsistent pair of linear equations.

How do you tell if an equation has no solution?

The coefficients are the numbers alongside the variables. The constants are the numbers alone with no variables. If the coefficients are the same on both sides then the sides will not equal, therefore no solutions will occur. Use distributive property on the right side first.

How do you know if a system has no solution?

If a system has no solution, it is said to be inconsistent . The graphs of the lines do not intersect, so the graphs are parallel and there is no solution.

When two lines are parallel the system has an infinite number of solutions?

When the lines are parallel, there are no solutions, and sometimes the two equations will graph as the same line, in which case we have an infinite number of solutions. Some special terms are sometimes used to describe these kinds of systems.

What is a unique equation?

By the term unique solution, one mean to say that only one specific solution set exists for a given equation. ... So, if we have two equations, then unique solution will mean that there is one and only point at which the two equations intersect.

How do you identify a 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.

How do you solve a linear equation that is homogeneous?

Use Gaussian elimination to solve the following homogeneous system of equations.
  1. Solution: By elementary transformations, the coefficient matrix can be reduced to the row echelon form.
  2. Solution check: Show that the set of values of the unknowns.
  3. Solution: Transform the coefficient matrix to the row echelon form:

What is homogeneous and non homogeneous?

A homogeneous system of linear equations is one in which all of the constant terms are zero. A homogeneous system always has at least one solution, namely the zero vector. ... A nonhomogeneous system has an associated homogeneous system, which you get by replacing the constant term in each equation with zero.

What does a determinant of 0 mean?

When the determinant of a matrix is zero, the volume of the region with sides given by its columns or rows is zero, which means the matrix considered as a transformation takes the basis vectors into vectors that are linearly dependent and define 0 volume.

What is an example of no solution?

A system has no solution if the equations are inconsistent, they are contradictory. for example 2x+3y=10, 2x+3y=12 has no solution. is the rref form of the matrix for this system. ... The row of 0's only means that one of the original equations was redundant. The solution set would be exactly the same if it were removed.

What is an example of infinitely many solutions?

When a problem has infinite solutions, you'll end up with a statement that's true no matter what. For example: 3=3 This is true because we know 3 equals 3, and there's no variable in sight. Therefore we can conclude that the problem has infinite solutions. You can solve this as you would any other equation.