Test Questions on Homogeneous vs. Non-homogeneous Systems (Linear Algebra)

📚 Quick Study Guide

• 🔢 Homogeneous System: A system of linear equations is homogeneous if all the constant terms are zero. It can be represented as $Ax = 0$, where $A$ is the coefficient matrix and $x$ is the vector of unknowns.
• ✅ Trivial Solution: Every homogeneous system has at least one solution, the trivial solution ($x = 0$).
• 🧐 Non-Trivial Solutions: A homogeneous system has non-trivial solutions if and only if the determinant of $A$ is zero (if $A$ is a square matrix), or if the rank of $A$ is less than the number of unknowns.
• 📝 Non-Homogeneous System: A system of linear equations is non-homogeneous if at least one constant term is non-zero. It can be represented as $Ax = b$, where $b$ is a non-zero vector.
• 💡 Solution Existence: A non-homogeneous system $Ax = b$ has a solution if and only if $b$ is in the column space of $A$, or equivalently, if the rank of the augmented matrix $[A|b]$ is equal to the rank of $A$.
• 🔑 Unique Solution: A non-homogeneous system has a unique solution if the rank of $A$ equals the rank of $[A|b]$ and also equals the number of unknowns.
• ➕ Infinite Solutions: If the rank of $A$ equals the rank of $[A|b]$ but is less than the number of unknowns, the non-homogeneous system has infinitely many solutions.

Practice Quiz

1. Which of the following systems is homogeneous?

   1. $x + y = 1$, $x - y = 1$
   2. $2x + 3y = 0$, $x - y = 0$
   3. $x + y = 2$, $x - y = 0$
   4. $x = 1$, $y = 1$

2. A homogeneous system $Ax = 0$ always has:

   1. No solution
   2. A unique non-trivial solution
   3. The trivial solution
   4. Infinitely many solutions

3. For a non-homogeneous system $Ax = b$ to have a solution, what must be true about the ranks of $A$ and $[A|b]$?

   1. rank($A$) < rank($[A|b]$)
   2. rank($A$) > rank($[A|b]$)
   3. rank($A$) = rank($[A|b]$)
   4. rank($A$) ≠ rank($[A|b]$)

4. If a homogeneous system has more variables than equations, then it has:

   1. Only the trivial solution
   2. A unique solution
   3. No solution
   4. Infinitely many solutions

5. Which of the following is a non-homogeneous system?

   1. $x + y + z = 0$
   2. $2x - y = 0$
   3. $x + y = 5$
   4. $3x = 0$

6. If rank($A$) = rank($[A|b]$) = number of unknowns for the system $Ax=b$, then the system has:

   1. Infinitely many solutions
   2. No solution
   3. A unique solution
   4. At least two solutions

7. For a homogeneous system $Ax = 0$, if det($A$) ≠ 0, then the system has:

   1. Infinitely many solutions
   2. Only the trivial solution
   3. No solution
   4. A unique non-trivial solution