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Linear Algebra Glossary

Core terms with their geometric pictures — the vocabulary behind the linear-algebra visualizations.

Vectors

v Vector
An arrow with both a magnitude and a direction, written by its components. Equivalently, a point in space reached from the origin. Vectors add tip-to-tail and scale by stretching.
Components: v = (vₓ, v_y). The same arrow is "the displacement from the origin to that point."
‖v‖ Norm (magnitude)
The length of the vector — the hypotenuse of its components, by Pythagoras. It is the square root of the dot product of the vector with itself.
v̂ Unit vector
A vector of length 1 pointing the same way as v — direction stripped of magnitude. Obtained by dividing a vector by its own norm ("normalising"). Every unit vector lies on the unit circle.

Span, Independence & Basis

Σ cᵢvᵢ Linear combination
Scale some vectors and add them. Every output you can build from a set of vectors is a linear combination of them — the single most important operation in linear algebra.
The dashed parallelogram shows 1.4·u + 1.0·w reached by stretching then adding tip-to-tail.
span Span
The set of all linear combinations of a collection of vectors. Two independent vectors span the whole plane (blue); if they are parallel the span collapses to a single line.
indep. Linear independence
No vector in the set is a linear combination of the others — none is redundant. Independent vectors point in genuinely different directions, so they enlarge the span.
Dependent ⇔ they lie on one line ⇔ the parallelogram they form has zero area.
basis Basis & dimension
A minimal set of vectors that is both independent and spanning. Every vector then has unique coordinates in that basis. The dimension is the number of basis vectors.
Here p = 2e₁ + 1·e₂ — its coordinates (2, 1) read off the e₁, e₂ basis.

Dot Product & Angle

v·w Dot product
Multiply matching components and sum — a single number. Geometrically it measures how much of one vector points along the other (its projection, in orange). Zero means perpendicular.
θ Angle between vectors
The dot product, divided by the two lengths, gives the cosine of the angle between the vectors. This is how length-and-angle geometry falls out of pure component arithmetic.

Linear Transformations

A Matrix / linear map
A transformation that sends grid lines to grid lines, keeping the origin fixed and preserving evenly-spaced parallels. Its columns are where the basis vectors e₁, e₂ land — those two arrows fully determine the map.
Faint = original unit square; solid = its image. The columns of A are the edges of the transformed square.
det A Determinant
The factor by which the map scales area (volume in higher dimensions). A negative determinant flips orientation; a zero determinant squashes space onto a line — the map is then non-invertible.
tr A Trace
The sum of the diagonal entries. It is basis-independent and equals the sum of the eigenvalues. Geometrically it is the rate at which area grows under the flow generated by A.

Kernel & Range

V Domain
The space the transformation takes its inputs from — every vector x that can be fed in. The kernel is the part of the domain that maps to zero.
W Codomain
The space the outputs are allowed to land in. The range is the part of the codomain actually reached — it can be smaller than W. The map is onto (surjective) exactly when range = W.
For an m×n matrix: domain = ℝⁿ, codomain = ℝᵐ. Domain ≠ codomain in general.
ker A Kernel (null space)
All inputs the map sends to the zero vector — what the transformation "forgets". For a rank-deficient map this is a whole line (or plane) of vectors collapsing to the origin. Lives in the input space.
range Range (image / column space)
Every output the map can produce — the span of its columns. Its dimension is the rank. Lives in the output space. Rank–nullity ties the two together: dim ker + rank = number of input dimensions.

Eigenvectors & Orthogonality

λ, v Eigenvalue & eigenvector
An eigenvector is a special direction the map only stretches, never rotates; the eigenvalue λ is the stretch factor. The blue vector stays on its own line (eigenvector); the grey one is knocked off its line.
A⁻¹ Inverse
The map that undoes A — applying A then A⁻¹ returns every vector to where it started. It exists only when det A ≠ 0; a singular matrix has collapsed information that cannot be recovered.
Q Orthogonal matrix
A transformation whose columns are orthonormal — a rotation or reflection. It preserves all lengths and angles, so the unit circle maps to itself. Its transpose is its inverse.