Linear Algebra
1.48 min read

Exceptional Behavior: When Coefficients Vanish

Most linear equations behave predictably — they define a hyperplane. But two special cases arise when the equation degenerates.

First: if all coefficients are zero but the RHS is nonzero, we get 0=b0 = b with b≠0b \neq 0. This is a contradiction — no values of the variables can satisfy it. The solution set is the empty set ∅\emptyset.

Second: if all coefficients are zero and the RHS is zero, we get 0=00 = 0. This is a tautology — every tuple (x1,…,xn)(x_1, \ldots, x_n) satisfies it. The solution set is all of Rn\mathbb{R}^n.

These degenerate cases occur during Gaussian elimination when a row of the matrix becomes all zeros. Recognizing them is critical for determining how many solutions a system has.

Formal View

Definition 1.4 — Degenerate Cases
For the equation 0⋅x1+0⋅x2+⋯+0⋅xn=b0 \cdot x_1 + 0 \cdot x_2 + \cdots + 0 \cdot x_n = b: \begin{itemize} \item If b≠0b \neq 0: the solution set is ∅\emptyset (no solutions — contradiction). \item If b=0b = 0: the solution set is Rn\mathbb{R}^n (all tuples — tautology). \end{itemize}

Why This Matters

Degenerate cases reveal whether a system is inconsistent or under-determined.

  • During Gaussian elimination, a zero row with nonzero RHS immediately signals no solution
  • A zero row with zero RHS means the system has one fewer effective equation than expected
  • These cases determine whether engineering systems have solutions, are over-constrained, or have free parameters

Quiz

Question 1

What is the solution set of 0x1+0x2=50x_1 + 0x_2 = 5?

Question 2

The equation 0x1+0x2=00x_1 + 0x_2 = 0 is satisfied by every point in R2\mathbb{R}^2.

Common Mistakes

  • Forgetting that 0=00 = 0 is a tautology (provides no constraint) while 0=c0 = c for c≠0c \neq 0 is a contradiction.
  • During elimination, discarding zero rows without checking whether the RHS is also zero.