Answer to Question #298751 in Linear Algebra for Nado

Question

Let T:R^n--> R^m be a linear transformation and let ( v1,v2,....v3) be a linearly dependent set. Show that the set ( T (v1),T(v2),....T(vn))is also necessarily linearly dependent.

Answer 1

Solution

Since (v1, v2,...vn) is linearly dependent then there is a non-trivial combination of them that equals zero.i.e.

a1v1 +a2v2+ ... +anvn = 0

Where not all ai =0.

By linearity T(a1v1 +a2v2+ ... +anvn) = 0

It can be rewritten as;

a1T(v1) +a2T(v2) + ... + anT(vn) = 0

Hence ( T (v1),T(v2),....T(vn)) is also linearly dependent as (v1, v2,...vn)