Unformatted text preview: Notes on linear algebra (Monday 17th October, 2016, 23:10) page 191 (thus, again, entry by entry), and 6} defined by
6’ = (0,0,0,...). (It makes sense to think of infinite sequences as 1 x ecu-matrices; thus, the above
deflnltlons of -|—, . and O are prec1sely the rules we set for matrices.) Example 4.10. Let S be any set. Consider the set IRS oféf all maps from S to IR.
Then, IRS becomes a vector space, if we define —|—, - and O as follows: I If f E IRS and g E 1RS are two maps, then their sum f + g is deﬁned to be
the map from S to ]R that sends each s E S to f (s) + g (9). Thus, (f‘l‘g) (5) = f(5) +38) for everys E S. This is called pointwise addition (because it means that we add two maps by
adding their values at each point). I If f E R5 is a map and A is a number, then the map A f is defined to be the
map from S to IR that sends each s E S to A - f (5) Thus, (Af) (s) = A*f(s) for every 3 E S. ...
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