A matrix A defines a function T(x)=Ax

Domain/codomain examples

Additivity: A(u+v)=Au+Av

Homogeneity: A(cu)=c(Au)

Knowing the output for basis vectors determines the output for all vectors

m×n matrix: m rows, n columns

Inputs have length n; outputs have length m

Vectors x are typically n×1

Multiplication Ax produces an m×1 vector

A=[a_ij] where a_ij is row i, column j

In the standard basis, the j-th column of A equals Ae_j

Interpretation: each column shows where a unit axis vector lands after transformation

If x=Σ_{j=1..n} x_j e_j, then

Meaning: output is a weighted combination of the columns of A

Each output component is a dot product of a row of A with x

(Ax)_i = row_i(A) · x

Scale (stretch/shrink)

Rotate (in 2D/3D with special matrices)

Reflect

Shear

Project (onto a line/plane)

Change coordinate systems (basis changes)

Translation (moving everything by a fixed vector) is not linear

To include translations, use affine transformations and homogeneous coordinates

2D: columns show where (1,0) and (0,1) go

Parallelogram interpretation

Applying B then A: A(Bx)=(AB)x

So AB represents the combined transformation

Generally AB≠BA

Geometric reason: doing two transformations in different orders can yield different results

Identity matrix I: Ix=x (does nothing)

Inverse matrix A^{-1} (if it exists): A^{-1}(Ax)=x

If A is not invertible, information is lost (e.g., projection)

Scale each axis independently

diag(d1,...,dn) scales coordinate i by d_i

Uniform scaling by c: cI

Preserve lengths and angles

2D rotation by θ: [[cosθ, -sinθ],[sinθ, cosθ]]

Flip across a line/plane

Determinant often -1 in 2D/3D orthogonal reflections

Slide one axis proportional to another

Example in 2D: [[1, k],[0, 1]]

Map onto a subspace

Idempotent property: P^2=P

Q^TQ=I

Preserve dot products, lengths, angles (rigid motions without scaling)

A^T=A

Often represent energy/quadratic forms; have orthogonal eigenvectors in real case

In 2D: |det(A)| scales area

In 3D: |det(A)| scales volume

Sign indicates orientation (flip if negative)

A is invertible iff det(A)≠0

det(A)=0 implies collapse into a lower dimension (non-invertible)

Dimension of the output subspace (how many independent directions remain)

Max possible rank is min(m,n)

Set of vectors mapped to zero: {x: Ax=0}

Nontrivial null space implies loss of information

Rank-nullity theorem: rank(A)+nullity(A)=n (for m×n)

v≠0 is an eigenvector if Av=λv

λ is the eigenvalue (scaling factor along that direction)

Certain directions stay on their span; only scaled (and possibly flipped if λ<0)

Helps describe repeated application A^k

Stability analysis, differential equations, principal components, diagonalization

A linear transformation is abstract; the matrix depends on chosen bases

If P changes coordinates, transformed matrix often becomes P^{-1}AP

Similar matrices represent the same linear map in different bases

Simplify the matrix (diagonal, block-diagonal)

Reveal invariant subspaces and eigen-structure

Ax=b describes inputs x that map to a target b

Unique solution: invertible square A

No solution: b not in the column space of A

Infinite solutions: nontrivial null space

Gaussian elimination performs row operations to solve and understand structure

Pure matrices cannot represent translation as a linear map in R^n

Use (n+1)×(n+1) matrices to combine rotation/scale/shear with translation

Common in computer graphics and robotics

Understand where basis directions go

Predict deformation of shapes

Magnitude: scaling of area/volume

Sign: orientation flip

Whether the transformation collapses dimensions

Whether it preserves lengths/angles or has special spectral properties

Diagonal → scaling

Orthogonal → rotation/reflection

Upper triangular → stepwise dependency