• Definition (Linear Map): A map T : V → W is linear if
T (u + v) = T (u) + T (v),
T (λv) = λT (v)
for all u, v ∈ V and scalars λ. Key point: Verify linearity by checking these two
properties.
• Injective ⇐⇒ Null(T ) = {0}: To show T is injective, show T (v) = 0 =⇒ v = 0.
• Invertibility: For T : V → W finite-dimensional with dim V = dim W ,
T invertible ⇐⇒ T injective ⇐⇒ T surjective.
Key point: Proving injectivity often suffices for invertibility.
• Isomorphism & Dimension: Two finite-dimensional vector spaces are isomorphic if and only if they have the same dimension.
• Given bases v1 , . . . , vn of V and w1 , . . . , wm of W , each T ∈ L(V, W ) corresponds
to a matrix [T ] ∈ Fm×n . The map T 7→ [T ] is an isomorphism L(V, W ) ∼
= Fm×n .
• Over C, every linear operator T : V → V is unitarily upper-triangularizable (Schur’s
Theorem).
• Eigenvalues: λ is an eigenvalue of T if (T − λI) is not invertible. Equivalently,
∃v ̸= 0 : T (v) = λv.
• Existence (Complex Case): Every operator on a nonzero complex vector space
has at least one eigenvalue.
• Diagonalizability:
T is diagonalizable ⇐⇒ V can be expressed as a direct sum of its eigenspaces.
More explicitly, if T is diagonalizable and has eigenvalues λ1 , . . . , λk , then
V = E(λ1 , T ) (direct sum) E(λ2 , T ) (direct sum) · · · (direct sum) E(λk , T ).
If T has dim V distinct eigenvalues, it is diagonalizable.
(Note: “Direct sum” means every element in V can be uniquely written as a sum
of elements from each of these subspaces, and the only vector common to all is the
zero vector.)
• Minimal Polynomial: For a polynomial q, q(T ) = 0 iff q is a multiple of the
minimal polynomial of T . The minimal polynomial governs factors and nilpotent
parts.
• Nilpotent Operators:
T nilpotent ⇐⇒ the only eigenvalue is 0 ⇐⇒ minimal polynomial is z m .
Such an operator can be represented by a strictly upper-triangular matrix in some
basis.
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• Generalized Eigenvectors & Jordan Decomposition: Over C,
V = G(λ1 , T ) (direct sum) G(λ2 , T ) · · · (direct sum) G(λm , T ),
where each G(λi , T ) is a generalized eigenspace.
• Characteristic Polynomial: Over C,
pT (z) = (z − λ1 )d1 · · · (z − λm )dm .
Cayley-Hamilton Theorem: pT (T ) = 0.
• Inner Product: ⟨·, ·⟩ is linear in the first argument, conjugate symmetric, and
positive definite. Use it to define norms and check orthogonality.
• Orthonormality: Orthonormal sets are linearly independent. Every finite-dimensional
inner product space has an orthonormal basis (Gram-Schmidt).
• Orthogonal Projections: For a subspace U , V = U (direct sum) U ⊥ . The
orthogonal projection PU satisfies PU2 = PU , range(PU ) = U , and gives the best
approximation in U .
• Adjoint (T ∗ ): Defined by ⟨T v, w⟩ = ⟨v, T ∗ w⟩. Properties: (ST )∗ = T ∗ S ∗ and
(T ∗ )∗ = T .
• Null/Range Relations:
Null(T ∗ ) = (range(T ))⊥ ,
Range(T ∗ ) = (Null(T ))⊥ .
• Self-Adjoint: T = T ∗ ; all eigenvalues are real; T is orthogonally diagonalizable.
• Normal: T T ∗ = T ∗ T . Normal operators are diagonalizable with an orthonormal
basis. If T is normal, eigenvectors from distinct eigenvalues are orthogonal.
• Spectral Theorems:
Over R : T self-adjoint ⇐⇒ T is orthogonally diagonalizable.
Over C : T normal ⇐⇒ T is unitarily diagonalizable.
Tips
• Dimension Arguments: If dim V = dim W , showing injectivity or surjectivity
alone implies invertibility.
• Minimal/Characteristic Polynomials: Use them to deduce eigenvalues, nilpotency, and diagonalizability quickly.
• Orthonormal Bases: Simplify computations and proofs involving inner products
and adjoints.
• Normal & Self-Adjoint Operators: Diagonalizability with an orthonormal basis
is guaranteed. For self-adjoint operators, the spectrum is real.
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