Definition 3.1 Let V and W be (real) vector spaces. A transformation T : V → W is said to be linear if, for all u, v ∈ V and ∈ R, Tu v Tu Tv Tu Tu # # Sometimes, we will leave out the parenthesis and use the notation Tu instead of Tu. The definition says that a mapping from V into W is a linear transformation if it respects the basic operations in vector spaces, namely,
... [Show full abstract] addition and multiplication by a scalar. Examples of such transformations abound in mathematics. For instance, the so called projection from R 3 to R 2 , defined by Pv 1 , v 2 , v 3 : v 1 , v 2 , ∀ v 1 , v 2 , v 3 ∈ R 3 and some other well known transformations on the plane, like reflection in a line and rotation about a point, are linear transformations. These are geometric examples. Two very important examples of linear transformations between infinite dimensional spaces are the derivative and integration. More precisely, if D maps a continuously differentiable function on a, b onto its derivative, that is, if D : C 1 a, b → Ca, b is the derivative map, then Df g Df Dg , ∀ f, g ∈ C 1 a, b Df Df , ∀ f ∈ C 1 a, b , ∀ ∈ R and S : Ca, b → Ca, b defined by Sf x : a x ftdt , ∀ f ∈ Ca, b , with a ≤ x ≤ b is such that, for all f, g ∈ Ca, b and ∈ R, Sf gx a x ft gtdt a x ftdt a x gtdt and Sfx a x ftdt a x ftdt Two basic properties of linear transformations are obtained by putting 0 and −1 in (ref: tresdois): T0 0 and T−u −Tu , ∀ u ∈ V Observe that we are using the same symbol for the zero vector in the space V as well as W. It can be easily shown that the set of all linear transformations from V to W, denoted LV, W, is itself a (real) vector space, with addition and multiplication by a scalar defined by T Su : Tu Su , ∀T, S ∈ LV, W and u ∈ V Tu : Tu , ∀T ∈ LV, W , u ∈ V and ∈ R We can define an operation that takes each T, S ∈ LX, W LV, X into TS ∈ LV, W, called composition or product, by putting
