Determine the dimension of a b c d R4 d a b c a b Soluti

Determine the dimension of {(a, b, c, d) R^4: d = a + b, c = a - b}.

Solution

An element in this set is

(a,b,a-b,a+b)=a(1,0,1,1)+b(0,1,-1,1)

Hence it can be spanned by 2 vectors. We now check if the vectors are linearly independent

x(1,0,1,1)+y(0,1,-1,1)=0

Hence, x=y=0

HEnce there are linearly independent

Hence dimension is 2