1.1 Systems of Linear Equations · Linear Algebra Refresher

Everything downstream — determinants, rank, eigenvectors, least squares — is ultimately a question about a linear system. This section fixes the vocabulary and the one legal move set: the three elementary row operations.

A linear equation in $n$ unknowns is $a_{1} x_{1} + a_{2} x_{2} + \dots + a_{n} x_{n} = b$ . Stack $m$ of them and you have an $m \times n$ linear system:

a_{11} x_{1} + a_{12} x_{2} + \dots + a_{1 n} x_{n} a_{21} x_{1} + a_{22} x_{2} + \dots + a_{2 n} x_{n} a_{m 1} x_{1} + a_{m 2} x_{2} + \dots + a_{mn} x_{n} = b_{1} = b_{2} ⋮ = b_{m}

Definition · Consistency

A solution is an ordered

n

-tuple

(x_{1}, \dots, x_{n})

satisfying every equation. The system is consistent if at least one solution exists and inconsistent if none does. The set of all solutions is the solution set.

★Only three outcomes

For a

2 \times 2

system each equation is a line: the lines meet at a point (one solution), are parallel (none), or coincide (infinitely many). The same trichotomy holds for every

m \times n

system — exactly one, none, or infinitely many. If you ever produce 'exactly 2 solutions' on an exam, you've made an arithmetic error.

Equivalent systems

Definition

Two systems in the same variables are equivalent if they have the same solution set. Three operations always produce an equivalent system:

1Interchange two equations (rows).
2Multiply an equation by a nonzero real number.
3Add a multiple of one equation to another.

!Exam trap

Operation II fails if the scalar is

0

: you'd replace a real constraint by

0 = 0

and potentially gain solutions. That 'nonzero' is not decoration — it's exactly why these three operations are reversible, which is what makes the systems equivalent.

Triangular form and back substitution

Definition · Strict triangular form

A system is strictly triangular if in the

k

th equation the coefficients of the first

k - 1

variables are zero and the coefficient of

x_{k}

is nonzero. Equivalently: the coefficient matrix is upper triangular with no zeros on the diagonal.

Such a system unravels from the bottom: solve the last equation for $x_{n}$ , substitute upward, repeat. Every value is forced, so the solution is unique. The whole point of Gaussian elimination is to manufacture this shape.

Worked example

Elimination to triangular form

0/5 steps

Solve the system by row reducing the augmented matrix.

x_{1} + 2 x_{2} + x_{3} 3 x_{1} - x_{2} - 3 x_{3} 2 x_{1} + 3 x_{2} + x_{3} = 3 = - 1 = 4

→Shortcut

Choose your pivot to keep the arithmetic clean. If the top-left entry is

0

or ugly, swap rows first (operation I) — that's free and it's exactly what the algorithm allows.

Check your work

Row reduction / rank / null space

Exact fraction arithmetic — type entries like 3, -2, or 1/2. One row per line.

Matrix

1	2	-1	1
2	4	-3	0
1	2	1	5

→

1	2	0	3
0	0	1	2
0	0	0	0

rank

nullity

cols (n)

pivot cols

1, 3

Basis for N(A)

-2

-3

-2

Basis for the column space — original columns 1, 3

-1

-3

Row reduce anything and compare against your hand work step by step.

Check yourself

A system of

m

linear equations in

n

unknowns is found to have two distinct solutions. How many solutions does it have in total?

T / F

Multiplying one equation of a system by

0

produces an equivalent system.

T / F

If an

n \times n

system can be reduced to strict triangular form, it has exactly one solution.

1.1Systems of Linear Equations

What you must be able to do

Equivalent systems

Triangular form and back substitution

Elimination to triangular form

Row reduction / rank / null space

Check yourself