4 Let U be the set of all vectors u in R4 such that 2u1 3u3

4. Let U be the set of all vectors u in R^4 such that 2u1 + 3u3 - 2u4 = 0 (i.e. U is the solution space of given system). (a) Find the dimension of U. (b) Find a basis of U. (c) Write U using this basis.

Solution

a) dimension is 4 by 3 ,,,,, b/c u_i^s are in R ⁴ and there are three such vectors

b) basis is u1={1,-1,1,1} , u3={ 2,2,0,4} , u4={ 4,2,1,7}

c) U= c1 u1 + c3 u3 +c4 u4

where c1, c3, c4 are 4 dimensional vectors and U is linear combinations of u1,u3,u4 s.t it satisfies the equation