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Section
Vector spaces, linear maps, inner products, and spectral theory
Click any term to expand its definition inline.
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A vector space over a field is a set equipped with two operations (functions) and such that for all and :
Vector spaces are the basic objects studied via linear maps and invariants such as the determinant and eigenvalues of operators.
A Euclidean space is a real inner product space with .
A linear map is a function between vector spaces over the same field such that for all and ,
A linear operator is a linear map on a vector space .
An eigenvalue of a linear operator is a scalar such that there exists a nonzero vector with
A vector satisfying this equation is an eigenvector, and all such vectors (together with ) form the eigenspace for . Eigenvalues are precisely the roots of the characteristic polynomial.
An eigenvector of a linear operator is a nonzero vector for which there exists a scalar such that
The corresponding scalar is an eigenvalue of .
An eigenspace of a linear operator associated to a scalar is the set
A characteristic polynomial of a linear operator on an -dimensional vector space is the polynomial
where is the matrix of in any basis and is the identity matrix. This definition is independent of the chosen basis.
A minimal polynomial of a linear operator on a finite-dimensional vector space is the unique monic polynomial of least degree such that
meaning that substituting into the polynomial yields the zero operator.
A determinant is a function that assigns to each matrix over a field the scalar
where is the set of permutations of and is the sign of .
A trace is the scalar associated to an matrix over a field defined by
For a linear operator on a finite-dimensional vector space, the trace is defined as for any matrix representing in a basis; this is independent of the basis.
An inner product space is a vector space over or together with a specified inner product on .
Orthogonality is the relation in an inner product space defined by
A norm on a vector space over is a function such that for all and :
Here denotes the absolute value of the scalar (for or ).
A normed vector space is a vector space equipped with a norm on .
A Euclidean norm on a Euclidean space is the norm defined by
An operator norm of a linear map between normed vector spaces and is the quantity
with the understanding that the supremum may be in general. When , one says is bounded.
A Banach space is a normed vector space such that every Cauchy sequence in converges (in the norm) to a point of .
A Hilbert space is an inner product space that is complete with respect to the induced norm .
A compact operator is a linear operator between normed vector spaces such that the image of the closed unit ball
has compact closure in .
Basis existence theorem: Every vector space has a subset such that every can be written uniquely as a finite linear combination of elements of .
Rank–nullity theorem: Let be a linear map between finite-dimensional vector spaces. Define
Then
In particular, the rank and the nullity satisfy .
Cayley–Hamilton theorem: Let be a linear operator on a finite-dimensional vector space . Let be the characteristic polynomial of . Then
meaning that substituting into its own characteristic polynomial yields the zero operator on .
Cauchy–Schwarz inequality: In an inner product space , for all ,
where is the induced norm. Equality holds if and only if and are linearly dependent.