One of the lovely properties of a linear map T on a vector space like R (distinguishing it from other arbitrary functions) is that we only need to know what I does to a small number of linearly independent vectors to define the entire map. For example, consider a linear map T from R2 to R3, where we know 8 5 (1)-(;) and (9)-(3) = 2 T( = -5 5 -5 We can see that (1) Since we were promised I was linear, it is easy to work out the value of T( must be equal to the linear combination (3) = Using this idea we calculate that T (()) + T((^)) = 1) + Note: the Maple notation for the vector - (-) -- 2 3 is < 1,2,3 >. (9) (()) since this (9) T(