Complete the following definition the transformation T is li

Complete the following definition the transformation T is linear if ... Use part a to show that if T in a linear transformation, then T(0) = 0 T(cu + dv) = cT(u) + dt(v), for all vectors u, u e R^n and for all scalars

Solution

(a) The definition of alinear transformation is as under:

A linear transformation is a mapping of points from one dimension to another. The notation of a basic linear transformation is given by T : Rⁿ R^m .

Properties of Linear Transformations:

A transformation T : Rⁿ R^m is linear iff (if and only if) the following properties hold.

1. T(u+ v)   = T(u)+ T (v)

2. T (cu) = cT(u) , for any scalar c R

3. T (0) =0

(b) If T is a linear transformation, then by definition itself, T (0) = 0 (by the 3rd property)

T (cu + dv) = T (cu) + T (dv) ( by the 1st property)

or, T (cu + dv ) = c T(u) + d T (v) (by the 2nd property

for all vectors u, v Rⁿ and for all scalars c, d R