Linear Algebra: Vector Spaces, Transformations, and Matrices

Vector Space

A set V with vector addition and scalar multiplication satisfying closure, associativity, commutativity, identity, inverses, and distributive properties.

Subspace

A subset W of V such that 0∈W, closed under addition and scalar multiplication.

Linear Transformation

A map T:V→W satisfying T(x+y)=T(x)+T(y) and T(kx)=kT(x).

Source & Target

For T:V→W, V is the source (domain), W is the target (codomain).

Standard Matrix

Matrix A where Ax=T(x) for all x in R^n.

Transpose

AT has entries (i,j) = (j,i) of A.

Symmetric Matrix

A matrix where A^T = A.

Skew-Symmetric Matrix

A matrix where A^T = -A.

Diagonal Matrix

A(i,j)=0 for i≠j.

Upper Triangular Matrix

A(i,j)=0 for i>j.

Lower Triangular Matrix

A(i,j)=0 for i

Injective

f(x)=f(x') implies x=x'.

Surjective

For all y in Y, ∃x in X s.t. f(x)=y.

Bijective

f is both injective and surjective.

Isomorphism

A bijective linear transformation between vector spaces.

Inverse Mapping

For bijective f:X→Y, f⁻¹:Y→X satisfies f∘f⁻¹=I and f⁻¹∘f=I.

Invertible Transformation

∃S s.t. S(T(v))=v and T(S(w))=w.

Invertible Matrix

A has inverse A⁻¹ satisfying AA⁻¹=A⁻¹A=I.

Kernel

ker(T)={v∈V | T(v)=0}.

Image

im(T)={T(v)|v∈V}.

Linear Combination

c₁v₁+...+cₙvₙ for scalars cᵢ.

Span

All linear combinations of a set of vectors.

Linear Relation

Equation c₁v₁+...+cₙvₙ=0.

Linearly Dependent

∃nonzero cᵢ such that Σcᵢvᵢ=0.

Linearly Independent

Σcᵢvᵢ=0 implies all cᵢ=0.

Basis

Linearly independent set that spans V.