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The Linear Algebra a Beginning Graduate Student Ought to Know

Jonathan S. Golan

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Institución detectada	Año de publicación	Navegá	Descargá	Solicitá
No detectada	2007	SpringerLink

Información

Tipo de recurso:

libros

ISBN impreso

978-1-4020-5494-5

ISBN electrónico

978-1-4020-5495-2

Editor responsable

Springer Nature

País de edición

Reino Unido

Fecha de publicación

2007

Información sobre derechos de publicación

Cobertura temática

Matemáticas

Tabla de contenidos

Verificá que desde tu institución tengas acceso para descargar o solicitar el libro completo o alguno de sus capítulos.

doi: 10.1007/978-1-4020-5495-2_11

Determinants

Jonathan S. Golan

Let F be a field and let n be a positive integer. We would like to find a function from M _nxn( F ) to F which will serve as an oracle of singularity, namely a function that will assign a value of 0 to singular matrices and a value other than 0 to nonsingular matrices.

Palabras clave: Positive Integer; Nonsingular Matrix; Elementary Matrice; Determinant Function; Block Form.

Pp. 199-228

doi: 10.1007/978-1-4020-5495-2_12

Eigenvalues and eigenvectors

Jonathan S. Golan

One of the central problems in linear algebra is this: given a vector space V finitely generated over a field F , and given an endomorphism a of V , is there a way to select a basis B of V so that the matrix Φ_BB(α) is as nice as possible? In this chapter we will begin by defining some basic notions which will help us address this problem.

Palabras clave: Positive Integer; Vector Space; Diagonal Matrix; Characteristic Polynomial; Canonical Basis.

Pp. 229-266

doi: 10.1007/978-1-4020-5495-2_13

Krylov subspaces

Jonathan S. Golan

Let V be a vector space over a field F and let α ∈ End(V) .

Palabras clave: Vector Space; Characteristic Polynomial; Canonical Basis; Krylov Subspace; Minimal Polynomial.

Pp. 267-284

doi: 10.1007/978-1-4020-5495-2_14

The dual space

Jonathan S. Golan

Let V be a vector space over a field F . A linear transformation from V to F (considered as a vector space over itself) is a linear functional on V .

Pp. 285-298

doi: 10.1007/978-1-4020-5495-2_15

Inner product spaces

Jonathan S. Golan

In this chapter, we will have to restrict the set of fields over which we work.

Pp. 299-324

doi: 10.1007/978-1-4020-5495-2_16

Orthogonality

Jonathan S. Golan

Let V be an inner product space and let 0_V ≠ v,w ∈ V . From Proposition 15.2 we see that

Pp. 325-348

doi: 10.1007/978-1-4020-5495-2_17

Selfadjoint Endomorphisms

Jonathan S. Golan

Let V be an inner product space. An endomorphism α of V is selfadjoint if and only if 〈(v),α,w〉 = 〈v,α(w)〉 for all v,w ∈ V .

Palabras clave: Linear Transformation; Symmetric Matrix; Positive Real Number; Product Space; Symmetric Matrice.

Pp. 349-368

doi: 10.1007/978-1-4020-5495-2_18

Unitary and Normal endomorphisms

Jonathan S. Golan

Let V be an inner product space. An automorphism of V which is an isometry is called a unitary automorphism .

Pp. 369-388

doi: 10.1007/978-1-4020-5495-2_19

Moore-Penrose pseudoinverses

Jonathan S. Golan

Let V and W be inner product spaces, and let α : V → W be a linear transformation.

Palabras clave: Positive Integer; Linear Transformation; Identity Function; Product Space; Canonical Base.

Pp. 389-398

doi: 10.1007/978-1-4020-5495-2_20

Bilinear transformations and forms

Jonathan S. Golan

Let V, W , and Y be vector spaces over a field F . We say that a function f : V × W → Y is a bilinear transformation if and only if the function.

Palabras clave: Positive Integer; Vector Space; Quadratic Form; Tensor Product; Linear Transformation.

Pp. 399-422