suppose that AB is a basis for a vector space W Prove that X

suppose that {A,B} is a basis for a vector space W. Prove that {X,Y} is also a basis for W where X=2A+3B and Y=3A-5B.

Solution

A vector in span{X,Y} is

r(2A+3B)+s(3A-5B)=(2r+3s)A+(3r-5s)B

Hence it is also in span{A,B}

Consider a vector in span{A,B}

rA+sB

Let, rA+sB=m(2A+3B)+n(3A-5B)

Comparing coefficients gives

r=2m+3n

s=3m-5n

Multiplying first by 3 and second by 2 and subtracting gives\\\\

3r-2s=19n

n=(3r-2s)/19

Multiplying first by 5 and second by 3 and adding gives\\\\

5r+3s=19m

m=(5r+3s)/19

Hence the vector also lies in span{X,Y}

HEnce, span{X,Y}=span{A,B}=W