1.1410 min read

Matrix Formulation and Forward Look

A linear system of $m$ equations in $n$ unknowns can be compactly written as $A\mathbf{x} = \mathbf{b}$ , where $A$ is an $m \times n$ matrix (the coefficient matrix), $\mathbf{x}$ is a column vector of unknowns, and $\mathbf{b}$ is the right-hand side vector.

Gaussian elimination corresponds to multiplying $A$ on the left by elementary matrices — one for each row operation. The sequence of operations can be packaged into an LU decomposition: $A = LU$ , where $L$ is lower triangular and $U$ is upper triangular (echelon form).

This matrix perspective opens the door to the deep theory of linear algebra: column spaces, null spaces, rank, invertibility. We've been doing linear algebra all along — now we have the language to say it precisely.

In upcoming chapters we will study: vectors and linear combinations, the geometry of column and null spaces, rank and nullity, matrix multiplication, and invertibility. Everything connects back to the solution structure we've seen here.

Formal View

Definition 1.14 — Matrix Form

A linear system

\sum_j a_{ij} x_j = b_i

for

i = 1, \ldots, m

is equivalently written as

A\mathbf{x} = \mathbf{b}

where

A = (a_{ij})

is an

m \times n

matrix,

\mathbf{x} = (x_1, \ldots, x_n)^T

, and

\mathbf{b} = (b_1, \ldots, b_m)^T

Remark 1.14 — LU Decomposition Preview

Gaussian elimination factors

A = PA^{-1}LU

where

P

is a permutation matrix,

L

is unit lower triangular (from the elimination multipliers), and

U

is upper triangular (the echelon form). This factorization allows solving

A\mathbf{x} = \mathbf{b}

O(n^2)

after

O(n^3)

preprocessing.

Why This Matters

Matrix notation is the compact language that unlocks all of linear algebra's power.

Scientific computing uses matrix algebra for everything from weather simulation to genome analysis
Deep learning represents neural networks as sequences of matrix multiplications
Quantum mechanics uses matrix algebra (Hermitian operators, eigenvalues) as its mathematical language
Computer graphics transforms scene geometry via 4×4 matrix multiplications per vertex

Learning Resources

Matrices as linear transformations

3Blue1Brown

The geometric meaning of matrices — the bridge to deep linear algebra.

10 min

Introduction to matrices

Khan Academy

Matrix notation, entries, and dimensions.

12 min

Quiz

Question 1

A linear system of 3 equations in 4 unknowns can be written as $A\mathbf{x} = \mathbf{b}$ where $A$ has size:

Question 2

The matrix $A$ in $A\mathbf{x} = \mathbf{b}$ completely encodes all information about the linear system.

Common Mistakes

Writing $A$ with dimensions transposed — rows = equations ( $m$ ), columns = unknowns ( $n$ ).
Forgetting that $A\mathbf{x} = \mathbf{b}$ is shorthand for the full system, not just one equation.