Let v1 v2 be a basis for R Let A be an invertible n times

Let {v_1, v_2, ..., } be a basis for R\". Let A be an invertible n times n matrix. Show that {Av_1, Av_2, ..., Av_n} is a basis for R^n.

Solution

Rn has dimension n so, v1,v2,...,vn form a linearly independent set since they span Rn

Now A is invertible hence kernel A={0}

Let, a1,...,an be real numbers so that

a1Av1+....+anAvn=0

A(a1v1+....+anvn)=0

But kernel ={0}

Hence,

a1v1+....+anvn=0

Now, v1,...,vn are linearly independent

Hence, a1=.....=an=0

Hence, Av1,...,Avn are linearly independent and hence span Rn and hence form basi for Rn