Geometric interpretation: arrows with magnitude and direction (in 2D/3D)

Abstract interpretation: elements of a vector space (can be higher-dimensional)

A basis is a set of independent vectors used to describe all vectors in the space

The same vector can have different coordinates under different bases

2D: e1 = (1, 0), e2 = (0, 1)

3D: e1 = (1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1)

A transformation T is linear if

A matrix A represents T in chosen bases such that

Multiplication A x produces a new vector that is the transformed version of x

For an n×n matrix A in the standard basis

Practical meaning

If A is m×n and x is n×1, then A x is m×1

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

A x is a weighted sum of the columns of A with weights from x

If A = [[a, b], [c, d]] and x = (x, y)

Uniform scaling: multiplies all lengths by the same factor

Non-uniform scaling: different factors along different axes

Diagonal matrices scale coordinate axes

Rotates vectors around the origin while preserving lengths and angles

Rotation matrices are orthogonal with determinant +1

Flips vectors across a line (2D) or plane (3D)

Often orthogonal with determinant -1

Slides points parallel to an axis proportional to the other coordinate

Preserves area in 2D for certain shears but changes angles

Maps vectors onto a subspace (line/plane)

Loses information (non-invertible unless projecting onto full space)

Any linear transformation can be seen as a combination of basic geometric effects

Linear transformations always map 0 to 0

Lines remain lines (though may rotate/scale/shear)

Parallel lines remain parallel

Shapes may distort, but gridlines remain straight and evenly parameterized

Represents “do nothing” transformation: I x = x

Collapses all vectors to the zero vector

A is invertible if there exists A⁻¹ such that A⁻¹ A = I

Transformation is reversible (one-to-one and onto in equal dimensions)

Measures area/volume scaling factor of the transformation

Sign indicates orientation flip (negative implies reflection-like behavior)

Determinant 0 implies non-invertible (collapses dimension)

Dimension of the output subspace (image) the transformation can reach

Lower rank means the transformation collapses space into a lower dimension

Sum of diagonal entries

Related to eigenvalues and certain dynamical behaviors

Quantify how much vectors can be stretched by the transformation

A nonzero vector v such that A v = λ v

Direction preserved; only scaled (possibly flipped if λ < 0)

λ is the scaling factor along eigenvector directions

Reveal “natural axes” of the transformation

Useful for understanding repeated application (dynamics), stability, and diagonalization

The same linear transformation has different matrices in different bases

If P converts coordinates from new basis to old basis, then

This is a similarity transform

Similar matrices represent the same linear map under different coordinate systems

Map n-dimensional space to itself

Determinant, eigenvalues, invertibility are central

Map n-dimensional inputs to m-dimensional outputs

Common in data transformation, linear models, and dimensionality change

Often represent quadratic forms; have real eigenvalues and orthogonal eigenvectors

Preserve lengths and angles: Qᵀ Q = I

Represent rotations/reflections (no stretching)

Pure scaling along coordinate axes (in that basis)

Useful for solving systems; represent transformations with sequential dependence

Finding x such that applying transformation A yields b

Unique solution: A invertible (in square case)

No solution: b not in the image (column space) of A

Infinitely many: nontrivial null space (dimension collapse)

Column space: all reachable outputs A x

Null space: all inputs mapped to zero (A x = 0)

2D/3D transformations: rotate, scale, shear, project

Composition of transforms via matrix multiplication

Linear models: predictions as matrix-vector products

Feature transformations and embeddings

Coordinate transforms, stress/strain relations, linearized systems

Filtering and transforms (e.g., discrete transforms as matrix operations)

Transition matrices mapping probability vectors forward

If first apply B then apply A, overall is A(Bx) = (A B) x

Matrix multiplication is generally not commutative: A B ≠ B A

Product matrix encodes the combined effect of multiple linear steps