Example : in the vector space v.(r), let a, = (1, 2, 1); a, = (3, 1, 5); a, = (3,-4, 7). then prove 2a, + sa, then prove that the subspaces spanned by s = {a,,

Example : In the vector space V.(R), let a, = (1, 2, 1); a, = (3, 1, 5); a, = (3,-4, 7). Then prove 2a, + Sa, Then prOve that the subspaces spanned by S = {a,, a} and T = {a,, a,, a} are the same. Solution. Since the linear span L(T) of T is the set of LC of the vectors a,, a , thereforelet L(T)= fa,a, + a,a,+ a,a, | a,, a, a, e R (a,(1, 2, 1) + a,(3, 1, 5) + a,(3,4, 7) .(1) Again, let a, = b,a, +ba, (3-4, 7) = b,(1, 2, 1) + b,(3, 1, 5) = (b, + 3b, 2b, + b, b, + 5b,) (2) b, +3b, =3 2b, +b, -4b, =-3, b, = 2 b, +5b, =7 Substituting these values of b, and b, in (2), we get (34, 7) = -3(1, 2, 1) + 2(3, 1, 5) a,(3,4, 7) = -3a,(1, 2, 1) + 2a,(3, 1, 5) .(3) Now by (1) and (3). L(T) {a, (1, 2, 1)+ a,(3, 1, 5)- 3a,(1, 2, 1) + 2a,(3, 1, 5)) {(a, - 3a,) (1, 2, 1) + (a, + 2a,) (3, 1, 5) = {a(1, 2, 1) + b(3, 1, 5) | a, b e P) = {aa, + ba) = L(S) h AnoP e pnan 3