What coordinates describe the object?
Choose a basis.
Coordinates depend on the frame used to represent the underlying vector.
Side 96
Linear
Algebra
A study of vectors, transformations and systems of linear relations. Linear algebra turns many-variable problems into geometry and structure: directions, combinations, subspaces, matrices and eigenmodes.
vector→combination→transformation→structure→solution
06linear lenses
05system questions
05decomposition tools
96Side
Vectors describe quantities with several coordinated components.
They can represent position, force, measurements, model parameters or any object whose components can be added and scaled.
Which vectors can be built from others?
Linear combination.
Weighted sums define reachability inside a vector space.
What region can the set generate?
All combinations.
The span reveals the subspace generated by available directions.
Is any direction redundant?
Remove unnecessary vectors.
Independent vectors contribute genuinely new directions.
What is the smallest complete coordinate system?
Independent + spanning.
A basis provides a compact representation of the entire space.
A linear system asks for values satisfying several constraints at once.
Matrix notation exposes whether the system has one solution, many solutions or none.
A linear constraint.
Each equation restricts the possible solution set.
Organized coefficients.
Matrices encode many simultaneous linear relations compactly.
Simplify without changing solutions.
Row operations reveal pivots, free variables and inconsistency.
Independent information.
Rank measures how many independent constraints or output directions remain.
Inputs mapped to zero.
The null space describes degrees of freedom invisible to the transformation.
Best fit when exact solution fails.
Projection finds the closest attainable solution to an inconsistent system.
A matrix is best understood as an action, not a table.
Linear transformations rotate, stretch, project, shear or otherwise remap vectors while preserving addition and scalar multiplication.
Input
Start with a vector.
The transformation acts on coordinates representing an underlying object.
Map
Apply the matrix.
Matrix multiplication combines input directions into new output directions.
Image
What outputs are reachable?
The column space is the set of outputs the transformation can produce.
Inverse
Can the mapping be undone?
An invertible transformation preserves enough information to recover the input uniquely.
Composition
Transformations can be chained.
Matrix multiplication represents sequential linear operations.
Geometry extends beyond three dimensions.
Inner products, norms and projections allow distance, angle and orthogonality to generalize to high-dimensional spaces.
| Concept | Question | Role |
|---|---|---|
| Norm | How large is a vector? | Magnitude and distance |
| Dot product | How aligned are two vectors? | Angle and similarity |
| Orthogonality | Are directions perpendicular? | Independent geometric components |
| Projection | What part lies along a subspace? | Approximation and decomposition |
| Change of basis | Which coordinates make structure simplest? | Representation without changing object |
Eigenvectors reveal directions a transformation preserves.
Along an eigenvector, the transformation changes only scale, not direction.
Invariant direction.
The transformation sends the vector onto the same line.
Scale along that direction.
Magnitude and sign determine growth, shrinkage or reversal.
Choose coordinates aligned with natural modes.
A complex transformation can become independent scalar actions.
Decompose any matrix geometrically.
Singular value decomposition separates rotations from directional stretching.
Find dominant variation.
Eigenstructure underlies principal component methods and covariance analysis.
Natural patterns evolve independently.
Dynamic and physical systems often simplify when expressed in modal coordinates.
Linear algebra is the grammar of multidimensional science.
It connects geometry, data, physics, differential equations, optimization and machine learning through a common representation language.
Data
Rows and columns become vectors, spaces and transformations.
Physics
States, forces and coordinate changes are naturally vectorial.
Differential equations
Linear systems evolve through matrix-defined dynamics.
Machine learning
Embeddings, layers and optimization operate in high-dimensional spaces.
Numerical science
Large computational models often reduce to repeated linear solves.
Introduction to Linear AlgebraGilbert Strang · geometric structure
Linear Algebra Done Rightvector spaces and operators
Matrix Computationsnumerical linear algebra
Applied Linear Algebramodels, data and computation