william_williams
william_williams 4d ago โ€ข 10 views

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

Hey there! ๐Ÿ‘‹ Feeling a bit lost with homogeneous and non-homogeneous systems in linear algebra? Don't worry, I've got you covered! This quick study guide and practice quiz will help you nail down the key concepts. Let's get started! ๐Ÿค“
๐Ÿงฎ Mathematics

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connor.mccoy Dec 27, 2025

๐Ÿ“š Quick Study Guide

  • ๐Ÿ”ข 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.
  • โœ… Trivial Solution: Every homogeneous system has at least one solution, the trivial solution ($x = 0$).
  • ๐Ÿง 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.
  • ๐Ÿ“ 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.
  • ๐Ÿ’ก 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 $A$.
  • ๐Ÿ”‘ 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.
  • โž• 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($A$) โ‰  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($A$) โ‰  0, then the system has:

    1. Infinitely many solutions
    2. Only the trivial solution
    3. No solution
    4. A unique non-trivial solution
Click to see Answers
  1. B
  2. C
  3. C
  4. D
  5. C
  6. C
  7. B

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