University of Oxford mathematician Dr Tom Crawford introduces the concept of a Linear Transformation with a motivation for the definition and several worked examples.

The video begins with the formal mathematical definition of a linear transformation and how their properties relate to those of a vector space. There is also a discussion as to how linear maps ensure that the structure of a vector space is preserved. An alternative definition for a linear transformation is then introduced, which is often used to check whether or not a given map is indeed linear. We also see how a linear transformation must always map the zero vector in the starting space to the zero vector of the final space. A generalised application to a basis of a vector space is also briefly mentioned (this will be discussed further in the next video on the Rank-Nullity Theorem.) Several examples of linear maps are looked at in detail, including multiplication by a matrix, differentiation, and three explicit maps in R3. The final example re-introduces the concept of a direct sum (as seen in the previous video) and how it can be used to define a projection map.