Vector spaces and linear transformations
An attractive feature of representation theory is that it combines two strands of mainstream mathematics, namely group theory and linear algebra. For reference purposes, we gather the results from linear algebra concerning vector spaces, linear transformations and matrices which we shall use later. Most of the material will be familiar to you if you have taken a first course on linear algebra, so we omit the proofs. An exception occurs in the last section, where we deal with projections; here, we explain in detail how the results work, in case you have not come across projections before.
Let F be either (the set of real numbers) or (the set of complex numbers). A vector space over F