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**AkilMAI** **1. The problem statement, all variables and given/known data**

For each of the following subsets U of the vector space V I have to decide whether or not U is a subspace of V . In each case when U is a subspace, I also must find a

basis for U and state dim U:

(i) V = R^4; U = {x = (x1; x2; x3; x4) : 3x1 - x2 -2x3 + x4 = 0}:

(ii) V = R^3; U = {x = (x1; x2; x3) : x1^2 = x2^2 + 4*x3^2}

(iii) V = P3; U = {p in P3 : p'(0) = p(1)}

I have some questions for each problem...

**2. The attempt at a solution**

I need to know that my solutions are ok or not and if so, how to find the basis and dim of U.

(i) we take x,y in U=> x+y=0 and a(a scalar),a*x=a*0=0 ,so it is closed under addition and scalar multiplication,therefore it is a subspace.If my approach is correct how can I find basis U and dim U?

(ii)Here I did something similair,I wrote x1^2 = x2^2 + 4*x3^2 as x1^2 - x2^2 - 4*x3^2=0

=> x,y in U where x+y =0+0=0 and a*x=a*0=0.Again If my approach is correct how cand I find basis U and dim U?

(iii)take p and g two polyn from U=>(g+p)(1)=p(1)+g(1)=p'(0)+g'(0)=(g+p)(0) and (a*p)(1)=a*p(1)=a*p(0)=(a*p)(0).Same thing, If my approach is correct how cand I find basis U and dim U?

Thank you