GraphMath

Linear Equations

Matrix form, vector form, RREF and the three solution cases

When does A x = b have one solution, no solution or infinitely many solutions?

This chapter presents linear equation systems in matrix form and vector form, compares them with matrix multiplication, solves them using reduced row echelon form and develops the three cases: unique solution, no solution and infinitely many solutions.

Open chapter PDF Key ideas Chapter index Back to Linear Algebra

Key ideas

A linear system can be read in two complementary ways: as a column-combination problem and as a set of row constraints.

  • The equation A x = b asks whether b can be built as a linear combination of the columns of A
  • The augmented matrix [ A | b ] packages the whole system into one object that can be simplified by row operations
  • Reduced echelon form reveals whether a solution exists and whether variables are pivot or free
  • Unique, inconsistent and infinite-solution cases arise from the relation between the columns of A and the target vector b
  • In the infinite-solution case, every solution is a particular solution plus a homogeneous direction in null(A)

The chapter develops both algebraic and geometric viewpoints, then connects the infinite-solution case to the decomposition of the domain into row space and null space.

What does the equation A x = b really mean?

It means that b must be expressible as a linear combination of the columns of A, with coefficients given by x. In the row view, the same system becomes a set of dot-product constraints that the unknown vector x must satisfy.

Related chapters

  • Row reduction explains echelon form, pivots and the algorithm used here to solve [ A | b ]
  • Matrix multiplication introduces the column-wise viewpoint needed to interpret A x = b as a linear combination of columns
  • Linear transformations frames solving A x = b as asking which targets are reachable under the map A
  • Rank supports the distinction between full-rank and rank-deficient cases and helps explain solvability
  • The four subspaces develops the row-space and null-space structure that appears in the later pages of this chapter

Chapter contents

The PDF is a single document. The page links below are best-effort: most browsers support them, but some viewers may ignore the page hint.

Topic Pages
Matrix representation of equation system 1
Vector representation of equation system 2–3
Linear equations vs matrix multiplication 3
Solving [ A | b ] using RREF 3–4
Case 1: unique solution 5
Geometric example of unique solution in 2D 6–8
Case 2: no solution 9–10
Geometric example of no solution in 2D 11–12
Case 3: infinitely many solutions 13–16
Geometric example of infinitely many solutions in 2D 16–18
Summary for case 3 with infinite solutions 18–19
From infinite solutions case to domain of n×n square matrix A 19–20
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Why can row operations solve the system without changing the answer?

Because each elementary row operation preserves the set of solution vectors x. Swapping equations, scaling an equation by a nonzero number or replacing one equation by a linear combination of equations changes the form of the system, but not which vectors satisfy it.

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