Let S u1 u2 un be linearly independent subset of Rm and B e

Let S = {u_1, u_2, ...u_n}, be linearly independent subset of R^m and B elementof M_m times m (R) is invertible. Prove that the set L_B(S) = {Bu_1, B_u2, ..., B_un} is also linearly independent.

Solution

Let, a1,...,an so that

a1Bu1+....+anBun=0

B(a1u1+...+an un)=0

B is invertible so ker(B)={0}

So, a1u1+...+anun=0

Now since u1,...,un are linearly independent so

a1=...-=an=0

Hence, Bu1,...Bun are linearly independent