What is the Solution Space of a Homogeneous System?

S7-SA2-0430

Grade Level:

Class 12

AI/ML, Physics, Biotechnology, FinTech, EVs, Space Technology, Climate Science, Blockchain, Medicine, Engineering, Law, Economics

Definition

What is it?

The solution space of a homogeneous system is the set of all possible solutions to a system of linear equations where all the constant terms are zero. It always includes the trivial solution (all variables are zero) and can also include non-trivial solutions.

Simple Example

Quick Example

Imagine you have two friends, Rahul and Priya, who want to share some laddoos. If 'x' is the number of laddoos Rahul gets and 'y' is the number Priya gets, and the total number of laddoos to be distributed is 0 (meaning no laddoos are actually there), then the equation is x + y = 0. The solutions could be x=0, y=0 (no one gets laddoos), or x=1, y=-1 (which doesn't make sense for laddoos, but mathematically works). The 'solution space' is all pairs (x, y) that satisfy this, like (0,0), (1,-1), (-2,2), etc., forming a line.

Worked Example

Step-by-Step

Let's find the solution space for the homogeneous system:
Equation 1: x + 2y - z = 0
Equation 2: 2x - y + 3z = 0

Step 1: Write the augmented matrix for the system.
[ 1 2 -1 | 0 ]
[ 2 -1 3 | 0 ]
---
Step 2: Perform row operations to get it into row echelon form. Multiply Row 1 by -2 and add it to Row 2 (R2 -> R2 - 2R1).
[ 1 2 -1 | 0 ]
[ 0 -5 5 | 0 ]
---
Step 3: Simplify Row 2 by dividing by -5 (R2 -> R2 / -5).
[ 1 2 -1 | 0 ]
[ 0 1 -1 | 0 ]
---
Step 4: Now, convert back to equations.
From Row 2: y - z = 0 => y = z
From Row 1: x + 2y - z = 0
---
Step 5: Substitute y = z into the first equation.
x + 2(z) - z = 0
x + z = 0
x = -z
---
Step 6: Let z be a free variable, say 't' (where 't' can be any real number). Then:
x = -t
y = t
z = t
---
Answer: The solution space is given by the set of vectors (-t, t, t) where 't' is any real number. This means any multiple of the vector (-1, 1, 1) is a solution.

Why It Matters

Understanding solution spaces helps engineers design stable structures and predict system behaviors. In AI/ML, it's crucial for understanding how algorithms find optimal solutions or patterns in data. Future scientists and engineers use this to model everything from rocket trajectories to protein folding.

Common Mistakes

MISTAKE: Forgetting that a homogeneous system always has at least one solution (the trivial solution). | CORRECTION: Always remember that (0, 0, ..., 0) is a valid solution for any homogeneous system, even if it's the only one.

MISTAKE: Confusing homogeneous systems with non-homogeneous systems. | CORRECTION: A system is homogeneous ONLY if all the constant terms on the right side of the equals sign are zero. If even one constant is non-zero, it's non-homogeneous.

MISTAKE: Not correctly identifying free variables when solving. | CORRECTION: After reducing the matrix, variables corresponding to columns without leading ones (pivots) are free variables and can be assigned parameters like 't' or 's'.

Practice Questions

Try It Yourself

QUESTION: Is the system 3x + 2y = 0 and x - y = 5 homogeneous? | ANSWER: No, because the second equation has a non-zero constant term (5).

QUESTION: Find the trivial solution for the system: 2a - b + c = 0, a + 3b - 2c = 0. | ANSWER: a=0, b=0, c=0.

QUESTION: Find the solution space for the homogeneous system: x + y = 0, 2x + 2y = 0. | ANSWER: Let y = t (a free variable). From x + y = 0, we get x = -y = -t. So the solution space is (-t, t) where 't' is any real number.

MCQ

Quick Quiz

Which of the following is true about the solution space of a homogeneous system?

It never contains the trivial solution.

It always contains at least one non-trivial solution.

It always contains the trivial solution.

It is always empty.

The Correct Answer Is:

A homogeneous system, by definition, has all constant terms equal to zero, meaning if all variables are zero, the equations are satisfied. Thus, the trivial solution (all zeros) is always part of its solution space. It may or may not have non-trivial solutions.

Real World Connection

In the Real World

Imagine you're designing an electric vehicle (EV) and need to balance the current flow in different parts of its electrical circuit. Using Kirchhoff's laws, you often end up with a homogeneous system of equations. The 'solution space' helps engineers understand all possible ways the currents can balance out, ensuring the EV runs safely and efficiently without short circuits or power loss.

Key Vocabulary

Key Terms

HOMOGENEOUS SYSTEM: A system of linear equations where all constant terms are zero | TRIVIAL SOLUTION: The solution where all variables are zero (e.g., x=0, y=0) | NON-TRIVIAL SOLUTION: A solution where at least one variable is non-zero | FREE VARIABLE: A variable in a system of equations that can take on any value, leading to infinite solutions | SOLUTION SPACE: The set of all possible solutions to a system of equations.

What's Next

What to Learn Next

Now that you understand homogeneous systems, explore 'Non-Homogeneous Systems of Linear Equations'. You'll see how a small change in the constant terms can drastically change the solution space and its properties, which is vital for solving real-world problems.