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1.1 Equations and solution sets

Treat a linear system as a set of simultaneous conditions, classify its solution set carefully, and see why equation operations preserve that set.

Course contents

MATH1030: Linear algebra I

Rigorous linear algebra notes on systems, matrices, structure, and proof, with interaction used only where it clarifies the mathematics.

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All note collectionsSeries overview
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Chapter 1Systems of equations1 sections
1.11.1 Equations and solution setsCurrent section
Chapter 2Matrices and elimination4 sections
2.12.1 Matrix basics2.22.2 Augmented matrices and row operations2.32.3 Gaussian elimination and RREF2.42.4 Solution-set types
Chapter 3Matrix algebra2 sections
3.13.1 Matrix multiplication and identity matrices3.23.2 Transpose and special matrices
Chapter 4Solution structure1 sections
4.14.1 Homogeneous systems and null space
Chapter 5Invertibility1 sections
5.15.1 Invertible matrices
Chapter 6Vector spaces6 sections
6.16.1 Vector spaces6.26.2 Subspaces6.36.3 Linear combinations and span6.46.4 Linear dependence and independence6.56.5 Basis and dimension6.66.6 Column space, row space, and rank
Chapter 7Determinants3 sections
7.17.1 Determinants and cofactor expansion7.27.2 Row operations, products, and invertibility7.37.3 Transpose, column operations, and Cramer's rule
Chapter 8Eigenvalues and diagonalization3 sections
8.18.1 Eigenvalues, eigenvectors, and eigenspaces8.28.2 Diagonalization and similarity8.38.3 Characteristic polynomials and diagonalization tests
Chapter 9Inner products and orthogonality4 sections
9.19.1 Inner products, norms, and angles9.29.2 Orthogonal sets and orthonormal bases9.39.3 Gram-Schmidt orthogonalization9.49.4 Cauchy-Schwarz and triangle inequalities

When a system of linear equations is written down, the real object of interest is not the list of equations itself. It is the set of all tuples of numbers that make every equation true at the same time.

That solution set can be a single point, no point at all, or an infinite family of points. The course notes begin with this language because every later method, from substitution to elimination to matrices, row reduction, and null spaces, is really a way of describing the same set more efficiently.

What a solution set records

Definition

Solution set

A solution set is the collection of every number or vector that satisfies the whole system.

If the system has unknowns x1,x2,…,xnx_1, x_2, \ldots, x_n, then a solution is an ordered n-tuple (s1,s2,…,sn)(s_1, s_2, \ldots, s_n) such that every equation becomes a true statement after we substitute xi=six_i = s_i for each i.

That order matters. The tuple (2, 3) is not the same solution as (3, 2). Likewise, a tuple is not a set. We do not write {2, 3} when the intended solution is the ordered pair (2, 3).

The system itself can be read as an intersection of conditions:

  • the first equation cuts out the tuples that satisfy it;
  • the second equation cuts out the tuples that satisfy it;
  • the solution set is the intersection of all those individual solution sets.

This is why even one failed equation disqualifies a tuple.

Consistent or inconsistent

Definition

Consistent and inconsistent systems

A linear system is consistent if it has at least one solution.

Otherwise, it is inconsistent.

This is a very small definition, but it carries a lot of meaning.

  • A consistent system may have exactly one solution.
  • A consistent system may have infinitely many solutions.
  • An inconsistent system has no solutions, so its solution set is empty.

Later row-reduction arguments prove that these are the only possibilities.

Theorem

A linear system has only three possible kinds of solution set

For a linear system, the solution set is either:

  1. a singleton, so the system has a unique solution;
  2. the empty set, so the system is inconsistent;
  3. an infinite set, so the system has infinitely many solutions.

The theorem is not a guess. Later sections justify it by elimination and row-reduction, but the point is already visible here: once a free variable appears, there are infinitely many choices; once a contradiction appears, there are none.

A system with one solution

Worked example

A tiny system with one solution

Solve

x+y=4,x−y=0.x + y = 4, \qquad x - y = 0.

The second equation says x=yx = y. Substitute that into the first equation:

y+y=4.y + y = 4.

So 2y=42y = 4, hence y=2y = 2, and therefore x=2x = 2.

The solution set is

{(2,2)}.\{(2, 2)\}.

This is the simplest possible example of a consistent system: there is exactly one ordered pair that works.

A system with no solution

Worked example

A contradiction gives an empty solution set

Consider

x+y=1,x+y=3.x + y = 1, \qquad x + y = 3.

If a pair (x, y) satisfied both equations, then the same left-hand side would have to equal two different numbers. Subtracting the first equation from the second gives

0=2,0 = 2,

which is impossible.

So the system is inconsistent, and its solution set is empty:

∅.\varnothing.

The important subtlety is that an inconsistent system is not a system with "many" solutions. It has none.

Common mistake

Inconsistent does not mean more than one solution

Students sometimes read "inconsistent" as "too complicated" or "overdetermined." That is not the definition. Inconsistent means that there is no tuple of values that satisfies all equations simultaneously.

A system with infinitely many solutions

Worked example

The same line can be written in more than one way

Consider

x+y=4,2x+2y=8.x + y = 4, \qquad 2x + 2y = 8.

The second equation is just 2 times the first, so it adds no new condition. Every pair on the line x+y=4x + y = 4 satisfies both equations.

If we let x=tx = t, then y=4−ty = 4 - t, where t∈Rt \in R. So the solution set is

{(t,4−t)∣t∈R}.\{(t, 4 - t) \mid t \in R\}.

There are infinitely many solutions because every real value of t gives a different ordered pair.

The same phenomenon appears in larger systems. The notation gets longer, but the logic is the same: each free variable introduces one independent choice.

Worked example

A four-variable system written as a full solution set

Solve

x1−2x2−x3+x4=1,x2+x3−x4=2,x3+2x4=3.x_1 - 2x_2 - x_3 + x_4 = 1, \qquad x_2 + x_3 - x_4 = 2, \qquad x_3 + 2x_4 = 3.

Start from the last equation:

x3=3−2x4.x_3 = 3 - 2x_4.

Substitute this into the second equation:

x2+(3−2x4)−x4=2,x_2 + (3 - 2x_4) - x_4 = 2,

so

x2=−1+3x4.x_2 = -1 + 3x_4.

Now substitute both expressions into the first equation:

x1−2(−1+3x4)−(3−2x4)+x4=1.x_1 - 2(-1 + 3x_4) - (3 - 2x_4) + x_4 = 1.

Simplifying gives

x1=2+3x4.x_1 = 2 + 3x_4.

Let x4=tx_4 = t. Then every solution has the form

(x1,x2,x3,x4)=(2+3t,−1+3t,3−2t,t),(x_1, x_2, x_3, x_4) = (2 + 3t, -1 + 3t, 3 - 2t, t),

so the solution set is

{(2+3t,−1+3t,3−2t,t)∣t∈R}.\{(2 + 3t, -1 + 3t, 3 - 2t, t) \mid t \in R\}.

This is the kind of answer the course wants: a complete description of all solutions, not just one sample solution.

Why equivalent systems matter

Two systems are equivalent if they have the same solution set.

Definition

Equivalent systems

Two linear systems are equivalent if and only if they have exactly the same solution set.

This definition is stronger than "they look similar" and stronger than "they have the same number of equations." Equivalence is about solutions only.

Common mistake

Same number of equations does not mean equivalent

Two systems can have the same number of equations but different solution sets. They can also have different numbers of equations and still be equivalent.

There are three elementary equation operations:

  1. swap two equations;
  2. multiply one equation by a nonzero scalar;
  3. add a multiple of one equation to another equation.

These are the equation-level version of the row operations used later on augmented matrices.

Theorem

Elementary equation operations preserve the solution set

If one system is obtained from another by a finite sequence of the three elementary equation operations, then the two systems are equivalent.

Proof

Why the three elementary equation operations are safe

This is the formal reason elimination is allowed. We are not changing the problem; we are rewriting it in a more readable form.

Two variables: geometry gives a quick picture

When there are two unknowns, each equation can be drawn as a line in the plane. Then the solution set is the intersection of those lines.

  • If the lines meet at one point, the system has a unique solution.
  • If the lines are parallel and distinct, the system is inconsistent.
  • If the lines coincide, the system has infinitely many solutions.

This geometric picture is useful because it makes the three possibilities feel inevitable instead of arbitrary.

Why the course starts with solution sets

The later matrix language does not replace this section. It formalizes it.

Once we introduce coefficient matrices and augmented matrices, the same system can be encoded more compactly. Once we introduce row operations, we can transform one equation system into an equivalent one. Once we reach row-echelon or reduced row-echelon form, the solution set becomes easier to read.

So if the solution set is the object, then every later technique is just a different lens.

Try the interactive preview below once you are comfortable reading a small system as a list of conditions.

Read and try

Translate one system into a matrix

The live explorer highlights how each equation becomes one matrix row plus one constant entry.

System

  1. x + 2y = 5
  2. 3x - y = 4

Result

125
3-14

Common mistakes and subtle points

Common mistake

A solution is an ordered tuple, not a bag of numbers

(2, 3) solves a system in two variables, but {2, 3} does not mean the same thing. Order matters because the first number belongs to x1x_1 and the second belongs to x2x_2.

Common mistake

A contradiction row means no solution

If elimination produces a row like 0=10 = 1, the system is inconsistent. Do not continue trying to solve it as if it were a valid equation.

Common mistake

A free variable is part of the answer

When a solution set is written with parameters, the parameter is not a missing answer. It is the correct way to describe the whole family of solutions.

Quick checks

Quick check

Which ordered pair solves both equations x+y=4x + y = 4 and x−y=0x - y = 0?

Test the pair against both equations.

Solution

Answer

Quick check

Is the system x+y=1x + y = 1, x+y=3x + y = 3 consistent?

Use the definition of consistency, not the number of equations.

Solution

Answer

Quick check

If two systems have the same solution set, what do we call them?

This is the course definition.

Solution

Answer

Quick check

Why does swapping two equations not change the solution set?

Think about what the word "solution" means.

Solution

Guided solution

Exercises

Quick check

Write the solution set of x1−5x4=1x_1 - 5x_4 = 1, x2+x4=2x_2 + x_4 = 2, x3+3x4=3x_3 + 3x_4 = 3 in parametric form.

Use one free parameter.

Solution

Guided solution

Quick check

Find c so that the system x+2y−5z=6x + 2y - 5z = 6, 2x+3y−2z=72x + 3y - 2z = 7, x+cy+z=0x + cy + z = 0 has no solution.

You may eliminate x first.

Solution

Guided solution

Read this first

This note is the starting point for the matrix treatment of systems. The next useful pages are:

Prerequisites

This section can be read on its own.

Key terms in this unit

More notes in this series

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2.1 Matrix basics