# Suppose that V and W are vector spaces over F. By a linear transformation from V to W, we mean a… 1 answer below »

Definition. Suppose that V and W are vector spaces over F. By a linear transformation from V to W,

we mean a mapping T : V ? W satisfying the following conditions:

(LT1) For every x, y ? V , we have T(x + y) = T(x) + T(y).

(LT2) For every c ? F and x ? V , we have T(cx) = cT(x).

Definition. Suppose that V is a vector space over F. A linear transformation T : V ? V is called a

linear operator on V .

Remark. Note that in the special case when W = F, a linear transformation T : V ? F is simply a

linear functional on V .

Definition. Suppose that V and W are normed vector spaces over F. Then a linear transformation

T : V ? W is said to be bounded if there exists a real number M = 0 such that kT(x)k = Mkxk for

every x ? V .

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