Consider the plane P in R4 containing all vectors of the for

Consider the plane P in R^4 containing all vectors of the form { a+b, a, 3b, b}, as a and b range over real numbers. Since this plane contains the origin and we know that any plan containing the origin is a subspace, P is a subspace of R^4. Find a basis for P

Solution

{a+b,a,3b,b}={a,a,0,0}+{b,0,3b,b}=a{1,1,0,0}+b{1,0,3,1}

HEnce set is spanned by

{(1,1,0,0),(1,0,3,1)}

To show that this also forms a basis we need to show they are linearly independent

Let,a,b so that

a(1,1,0,0)+b(1,0,3,1)=0

a+b=0

a=0,

3b=0

HEnce, a=b=0

And hence these two vectors span P and are linearly independent. Hence form basis for P