Question

1. Let S = {(a, b, c, d) e R4: a+b+c= 0} a). Show that S is a subspace of R4. b). Find a basis of S. 2. Let M = {(ui, uz, u3) € R3: U1 + U2 = 2). Is Ma subspace of R3? Explain your answer, if your answer is yes, give a proof why it is a subspace. If your answer is no, then show why it is not a subspace.

Answer 1

The ze 29) M= {(ui, U2, U3) € R’ | u.+ 12 = 2} IF M is a subspace of R3, then must contain the zero vector.. e zero vector in R3 is (0,0,0) and (0,0,0) & M oo 0+0+2 :: M is not a subspace of RS 19= fra, b, c, d)}E R4 | a+b+c=0} To show S is a subspace of IRS (0,0,0,0) ES % 0+0+0 = 0 (1) OES (2) Let X=(a, b, c, d) ES and, a+b+c= 0 - (I) 22=(,9,7, S) ES P+q7r = 0 -(II) Then x + x2 = (a+P; b+q, Ctr, d+s) Now a+P+b+q++r = atbtct pratr = 0 +0 -(from I SI) ós 21+X2 ES 3.) Letza, b, c, d) € s , at b+C =0 het & be a scalar * X(a, b, c, d) = (xa, xb, xc, xd) Now xatxbt ac = X(a+b+c) = 100)=0 "XXES i From (1) (2) $(3) Sis a subspace of R3

For Basis; Let (a,b,c,d)ES i atb+c=0 : a=-6-c Let be to, Ct2 a = -ti-tz Let d= t3 Then (a, b, c, d ) = (-ti-ta, ti, tz, t3) = tisti, ti (1,1) 0;o)+ t261, 0,1,0) +43 (0,0,0, .: Basis = {E1,1,0,0), (-1,0,1,0), (0,0,0,1)