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Interactive Audio Lesson
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Introduction to Consistency of a Linear System
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Today, we'll discuss how to determine if a linear system has solutions based on the consistency defined through the ranks of matrices. Can someone tell me what the term 'linear system' means?
A linear system consists of equations where the variables are raised only to the first power.
Exactly! Now, can you explain why we need to check for consistency?
We need to know if the equations can actually work together to find a solution.
Correct! That's where the rank comes in. Remember, if rank(A) equals rank([A∨B]), we have at least one solution.
Understanding the Rouché–Capelli Theorem
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Let's delve into the Rouché–Capelli theorem. Who can state the theorem in their own words?
The theorem says a system of linear equations is consistent if and only if the rank of the coefficient matrix equals the rank of the augmented matrix.
Fantastic! And if we find these ranks to be different?
Then the system has no solutions; it’s inconsistent.
Exactly! Can anyone summarize what happens if the ranks are equal or if one is less than the other?
If they’re equal and equal to the number of variables, we have a unique solution. If they’re equal but less, we have infinitely many solutions.
Example of a Consistent System
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Now, let's apply what we've learned. Consider the system: x + y + z = 6, x + 2y + 3z = 14, 2x + 3y + 4z = 20. Can we write the augmented matrix for this?
Sure! The augmented matrix would be [1 1 1 | 6; 1 2 3 | 14; 2 3 4 | 20].
Great job! Now, what do we do next?
We apply row operations to reduce it to row echelon form.
Correct! After reduction, what did you find?
We found rank(A) equals rank([A∨B]) equals 2, which means there are infinitely many solutions since it’s less than the number of variables.
Excellent! You’ve applied the concepts beautifully.
Introduction & Overview
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Quick Overview
Standard
In this section, the relationship between the ranks of the coefficient matrix and the augmented matrix is explored to understand the conditions under which a linear system has solutions. This includes the application of the Rouché–Capelli theorem and examples illustrating consistent and inconsistent systems.
Detailed
In linear algebra, the consistency of a linear system of equations is crucial for determining whether solutions exist. A system is expressed as AX = B, where A is the coefficient matrix, X is the vector of unknowns, and B is the constants vector. The consistency and potential solution forms depend on comparing the ranks of A and the augmented matrix [A∨B]. According to the Rouché–Capelli theorem, a system is consistent (has at least one solution) if and only if the rank of A equals the rank of [A∨B]. Furthermore, unique solutions arise when rank(A) equals the number of variables, whereas infinitely many solutions occur when rank(A) is less than the number of variables. Conversely, if the ranks differ, the system is inconsistent (no solutions). An example is provided to illustrate the process of determining the rank and identifying the nature of the solutions.
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Understanding the Linear System
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Let us consider a linear system of equations:
AX=B
Where:
- A is the coefficient matrix,
- X is the column vector of unknowns, and
- B is the column vector of constants.
Detailed Explanation
The equation AX = B represents a system of linear equations, where:
- 'A' represents the coefficients associated with the variables in your equations. For example, in the equation x + y = 3, 'A' would be [1, 1].
- 'X' is the vector that contains the variables (unknowns) that we are trying to solve for, such as [x, y].
- 'B' represents the constants on the right-hand side of the equations, which are the results of the equations. In our example, 'B' would be [3]. This setup allows us to manipulate the equations using matrix operations.
Examples & Analogies
Imagine you're trying to distribute some resources, like drinks among friends during a party. If A represents how much each friend wants (like their drink preferences), X represents the number of drinks you need to allocate to each friend, and B represents the total number of drinks you have. You need to figure out how to match these preferences with the available drinks, just like we solve for 'X' in the equation.
The Augmented Matrix
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The augmented matrix is
[A∨B].
Detailed Explanation
The augmented matrix combines the coefficient matrix 'A' and the constant vector 'B' into a single matrix representation. This matrix allows for easier manipulation and helps us to apply row operations to analyze the system as a whole. It visually represents both the coefficients of the unknowns and the constants that the equations equal to.
Examples & Analogies
Think of the augmented matrix as a complete list of ingredients and requirements for a recipe. For example, if you're making a cake, the matrix shows both the ingredients (coefficients) needed for each layer of the cake, while the constants tell you how many cakes you want to make.
Evaluating Consistency via Ranks
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The consistency and solution nature of this system depends on comparing the ranks of A and [A∨B].
Detailed Explanation
To determine if the system of equations is consistent (meaning it has at least one solution), we compare the rank of matrix 'A' (the coefficient matrix) with the rank of the augmented matrix [A∨B]. The rank of a matrix is essentially the maximum number of linearly independent rows or columns within it. If both ranks are equal, the linear system is consistent. If they are not equal, the system is inconsistent, meaning there are no solutions.
Examples & Analogies
Imagine you are a teacher trying to assign projects to students. If the number of projects (the rank of 'A') matches the number of available project outlines (the rank of [A∨B]), then all students can get a project. However, if you have more students than available outlines or if some outlines don’t match any project, you won’t be able to assign projects, similar to how a system becomes inconsistent.
Rouché–Capelli Theorem
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Theorem: Rouché–Capelli Theorem
A system of linear equations is consistent (i.e., has at least one solution) if and only if
rank(A)=rank([A∨B])
If the system is consistent:
- Unique solution if rank(A)=number of variables
- Infinitely many solutions if rank(A)<number of variables
If the system is inconsistent:
rank(A)≠rank([A∨B])
Detailed Explanation
The Rouché–Capelli theorem provides a criterion for determining the consistency of a system of linear equations based on ranks. It specifies that a system is consistent only if the rank of the coefficient matrix 'A' is equal to the rank of the augmented matrix. If they are equal, the number of solutions can vary: if the rank equals the number of variables, there is a unique solution; however, if the rank is less than the number of variables, there are infinitely many solutions. Conversely, if the ranks are not equal, the system has no solutions.
Examples & Analogies
Consider a popular restaurant's reservation system. If the number of tables (A) matches the number of reservations made (B), everyone gets seated smoothly. If there are more tables than reservations (rank(A) < rank([A∨B])), some tables will remain empty, indicating available opportunities. If reservations exceed tables (rank(A) ≠ rank([A∨B])), some customers will have to be turned away, representing an inconsistent situation.
Example of a Consistent System
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Example 1: Consistent System
Solve:
$$x + y + z = 6 \ x + 2y + 3z = 14 \ 2x + 3y + 4z = 20$$
Coefficient matrix A:
[1 1 1] [ 6 ]
1 2 3 B= 14
2 3 4 20
Form augmented matrix and apply row operations:
[1 1 1 6 ]
[A∨B]= 1 2 3 14 ⇒Reduce to REF
2 3 4 20
After reduction:
[1 1 1 6]
0 1 2 8 ⇒rank(A)=rank([A∨B])=2<3
0 0 0 0
⇒ Infinitely many solutions
Detailed Explanation
In this example, we have a system of equations that we solve step by step. The augmented matrix is formed by combining the coefficients and constants into a single matrix. After performing row operations to reduce it to Row Echelon Form (REF), we observe that there are two non-zero rows, indicating that the rank of matrix A is 2. Since the rank of [A∨B] is also 2 and is less than the number of variables (3), it shows there are infinitely many solutions. These solutions make sense as there’s not a single answer that satisfies all equations, but rather many combinations that work.
Examples & Analogies
Imagine three friends trying to match their schedules for a weekend trip. If they have overlapping times for different plans (such as camping, hiking, or home), there is not one set way to heroically solve who can do what, just many options and alternatives available. This scenario mimics our equations that yield infinitely many solutions: we can have many combinations of activities, as long as their preferred times overlap.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Rank of a matrix: It measures the maximum number of linearly independent row or column vectors.
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Augmented matrix: This is formed by combining the coefficient matrix and the constants of the linear equations.
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Rouché–Capelli Theorem: This theorem states that a linear system has a solution if the ranks of the coefficient matrix and augmented matrix are equal.
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Consistent and inconsistent systems: A consistent system has at least one solution, while an inconsistent system has no solutions.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Example of a consistent system: x + y + z = 6, x + 2y + 3z = 14, 2x + 3y + 4z = 20 leads to infinitely many solutions.
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Example of inconsistency: The system x + y = 1 and x + y = 2 has no solution.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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If rank’s not the same, solutions won’t claim.
📖 Fascinating Stories
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Imagine two friends trying to meet at a point. If they take different routes (ranks differ), they won't meet (no solution). If they stay on the same road (ranks equal), they'll find a way (solutions).
🧠 Other Memory Gems
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Remember to check Ranks: If they match, solutions hatch. If they diss, solutions miss.
🎯 Super Acronyms
C.R.A.N.
- Consistent Rank Augmented 'N' equations (C.R.A.N. = Consistent results when ranks agree).
Flash Cards
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Glossary of Terms
Review the Definitions for terms.
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Term: Rank
Definition:
The maximum number of linearly independent rows or columns in a matrix.
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Term: Augmented Matrix
Definition:
A matrix that is formed by appending the columns of the coefficient matrix and the constant terms.
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Term: Rouché–Capelli Theorem
Definition:
A theorem that describes conditions under which a system of linear equations has solutions based on the equality of ranks.
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Term: Consistent System
Definition:
A linear system that has at least one solution.
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Term: Inconsistent System
Definition:
A linear system that has no solutions.