Published online by Cambridge University Press: 22 September 2009
A linear algebra over the field of real numbers R is, by definition, a linear space A over R together with a bilinear map A2 → A, the algebra product.
Examples include R itself, the field of complex numbers, C, consisting of the linear space R2 with the product (a, b)(c, d) = (ac − bd, ad + bc), the double field2R consisting of the linear space R2 with the product (a, b)(c, d) = (ac, bd), and the full matrix algebraR(n) of all n × n matrices with real entries, with matrix multiplication as the product.
An algebra A may, or may not, have a unit element, and the product need be neither commutative nor associative, though it is usual to mention explicitly any failure of associativity. The unit element, if it exists, will normally be denoted by 1(A) or simply, where no confusion need arise, by 1, the map R → A, λ ↦ λ1(A) being injective. (The notation 1A is reserved for the identity map on A.)
All the above examples are associative and have a unit element, and all are commutative, with the exception of the matrix algebra R(n), with n > 1. The double field 2R is often identified with the subalgebra of R(2) consisting of the diagonal 2 × 2 matrices, the unit element being denoted by 21.
To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.