A system with fewer equations than unknowns is an underdetermined system, which has an infinite number of possible solutions.
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An underdetermined system is a system with fewer equations than unknowns, which means that there are an infinite number of possible solutions. This type of system is commonly found in real-life scenarios where not all the necessary information is available. One example is a system of linear equations in which there are more variables than equations.
According to the mathematician, Terence Tao, “An underdetermined system is one in which there are more unknowns than equations, so there are many possible solutions.”
Here are some interesting facts about underdetermined systems:
- Underdetermined systems are used in signal processing to reconstruct missing or incomplete data.
- These types of systems also arise in machine learning where finding a solution can be interpreted as finding a model that fits data.
- In some cases, finding a solution to an underdetermined system can be a difficult and complex problem.
- The rank of a matrix can indicate whether a system is underdetermined, overdetermined, or has a unique solution.
- Underdetermined systems can be solved using techniques such as the method of least squares, linear programming, or convex optimization.
Here is an example of an underdetermined system with two unknowns x and y and only one equation:
x + 2y = 3
The solution to this system is a line, and there are infinitely many points on that line that satisfy the equation. Therefore, there are infinitely many solutions to this underdetermined system.
An M x N matrix A is said to have:
- Full rank or rank N if its columns are linearly independent and span the N-dimensional space.
- Rank M<N if some rows are linear combinations of others. In this case, the number of pivots or leading 1’s in the row-reduced echelon form is less than M.
- Rank r if there is an r x r submatrix that has full rank but any (r + 1) x (r + 1) submatrix does not have full rank. In other words, the rank is the dimension of the row space or column space.
To summarize, an underdetermined system is a system with fewer equations than unknowns that has an infinite number of possible solutions. These types of systems are commonly found in real-life scenarios and can be solved using different techniques such as linear programming or least squares. The rank of a matrix can give an indication of whether a system is underdetermined or not.
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A system of polynomial equations which has fewer equations than unknowns is said to be underdetermined. It has either infinitely many complex solutions (or, more generally, solutions in an algebraically closed field) or is inconsistent.
In general, a system with fewer equations than unknowns has infinitely many solutions, but it may have no solution. Such a system is known as an underdetermined system . In general, a system with the same number of equations and unknowns has a single unique solution.
An underdetermined system is one with fewer equations than unknowns, so we can write this as a matrix equation A x = b with A a matrix that has fewer rows than columns. This implies that solutions, if they exist, will not be unique.
A system of linear equations with fewer equations than unknowns is sometimes called an underdetermined system.
Video answer to “What is a system with fewer equations than unknowns?”
This video showcases the process of using Gaussian elimination to solve a system of equations with fewer equations than variables. The instructor converts the system into a row echelon form by rewriting it as an augmented matrix, solves for one variable in terms of the others, and then substitutes to solve for the other variables. The solution for a system with two equations and three unknowns is determined to be 50 + Z for x, 10 – 2z for y, and Z for z.
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