Are you ready to dive deeper into the world of systems of equations? In this article, we will explore the intricacies of solving linear systems, focusing on their unique characteristics and different scenarios. Whether you’re a beginner or an experienced engineer, this guide will provide you with the knowledge you need to tackle these types of equations effectively.
Contents
Understanding Underdetermined Systems
Let’s start by examining a fascinating scenario: underdetermined systems. Imagine you’re given just one equation with three unknowns, such as Sally’s age plus Jon’s age plus Bob’s age equals 60. Can you determine the ages of Sally, John, and Bob? The answer is no. In an underdetermined system, there are infinitely many possibilities. For example, if Sally is 20, John could be 10 and Bob 30, or John could be 30 and Bob 10. The combinations are endless.
Working with Underdetermined Systems in MATLAB
Even though underdetermined systems lack enough information to yield a unique solution, MATLAB offers an interesting approach to tackle them. Let’s use the equation from the previous example: Sally’s age plus Jon’s age plus Bob’s age equals 60. We can represent this equation using matrices.
If we define matrix A as [1 1 1] and vector B as [60], we can solve for X (Sally, John, and Bob’s ages) using MATLAB’s backslash operator:
X = A BIn this case, MATLAB will output X = 60, indicating that Sally is 60 years old, while John and Bob haven’t been born yet or were just born. Since this is an underdetermined system, there are infinitely many solutions, and MATLAB provides one possibility.
The Pseudo Inverse: An Alternate Solution
Apart from the backslash operator, MATLAB also introduces the concept of the pseudo inverse. The pseudo inverse is useful for nonsquare matrices like the one we have in this underdetermined system. By utilizing the pseudo inverse and multiplying it by the vector B, we can obtain an additional solution for Sally, John, and Bob.
X = pinv(A) * BIn this case, MATLAB yields X = [20 20 20], suggesting that the best solution is for Sally, John, and Bob to all be 20 years old.
Overdetermined Systems: Too Many Constraints
Now, let’s explore another intriguing scenario: overdetermined systems. In an overdetermined system, we have more equations than unknowns. This situation can occur when we seek to solve for multiple unknowns with an excessive number of constraints. For instance, suppose we have the following system of equations:
- 2Sally + John – Bob = 50
- -Sally + John + Bob = 20
- Sally + John + 2Bob = 100
In this case, Sally, John, and Bob have specific values that satisfy all three equations. However, if we introduce another equation, such as Sally + John + 2Bob = 75, the system becomes overdetermined. The new equation contradicts the original ones, making it impossible to find a solution that satisfies all four equations simultaneously.
MATLAB’s Approach to Overdetermined Systems
Using MATLAB, we can attempt to solve overdetermined systems. However, it’s important to note that MATLAB will not find a unique solution in these cases. Instead, MATLAB will aim to provide an approximation that best satisfies the equations.
Let’s consider the overdetermined system mentioned earlier. By using the backslash operator in MATLAB, we can calculate the approximate solution:
X = A BIn this case, MATLAB suggests that the closest solution is Sally = 25.7, John = 20, and Bob = 24.3. Although these values are close, they do not perfectly satisfy all the equations. Similarly, if we apply the pseudo inverse, the result will be similar.
Avoiding Redundancy in Determined Systems
Lastly, we must address redundancy in determined systems. Even if we have three equations and three unknowns, we can encounter redundancy, ultimately resulting in no solutions. Redundancy occurs when one or more equations contradict each other.
For example, if we take the overdetermined system with three equations again:
- 2Sally + John – Bob = 50
- -Sally + John + Bob = 20
- Sally + John + 2Bob = 60
The first and third equations contradict each other since they cannot both hold true simultaneously. Consequently, this system has no solutions.
Conclusion
Solving systems of equations is a powerful tool in the field of mathematics and engineering. Whether you’re dealing with underdetermined, overdetermined, or determined systems, understanding their characteristics is crucial. MATLAB provides helpful functions, such as the backslash operator and the pseudo inverse, to handle these scenarios effectively.
Intrigued by the world of systems of equations? Explore more mathematical concepts and discover the endless possibilities they offer. For further resources and insights on technology and engineering, visit Techal.
FAQs
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Q: Can underdetermined systems ever have a unique solution?
- A: No, underdetermined systems lack enough information to yield a unique solution. They have infinitely many solutions.
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Q: Why do overdetermined systems not have unique solutions?
- A: Overdetermined systems have more equations than unknowns, leading to too many constraints. This excess of constraints makes it impossible to find a solution that satisfies all the equations.
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Q: Can determined systems have redundancy?
- A: Yes, determined systems can have redundancy if one or more equations contradict each other. Redundancy in determined systems results in no solutions.
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Q: What techniques can MATLAB use to solve underdetermined and overdetermined systems?
- A: For underdetermined systems, MATLAB can utilize the backslash operator and the pseudo inverse. However, it is essential to remember that these solutions are approximations.
