B

Background Mathematics

We begin with the definition of a vector space. Where appropriate, we will give simpler definitions, which at the expense of some generality will be sufficient for the use made of them in the text. For example a vector space can be defined over any field, but we will consider vector spaces over the real numbers, so that what we now introduce are sometimes called ‘real vector spaces’.

**Definition B.1** A set *X* is a *vector space (VS)* if two operations (addition, and multiplication by scalar) are defined on *X* such that, for **x**, **y** *X*, and ,

and such that in addition *X* is a commutative group with identity ...

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