Question

(1 point) Let u4 be a linear combination of {u1, U2, U3}. Select the best statement. O A. We only know that span{u1, U2, U3, u4} span{u1, u2, u3} . B. There is no obvious relationship between span{u1, U2, uz} and span{u1, U2, U3, u4} . C. span{u1, U2, U3} = span{u1, U2, U3, u4} when none of {u1, U2, uz} is a linear combination of the others. D. We only know that span{u1, U2, U3} C span{u1, U2, U3, u4} O E. span{u1, U2, u3} = span{u1, 42, 43, u4} . OF. none of the above

Answer 1

Let " be a linear combination of {41, 42, 433. de for some &, Bre field, we can get write, Uq au, + B + ru Now, span { 4, , %2, 43, 44} = { qui + Colt GM ₂ + Calle | 9, 2, 3, 4 e field] {94,+ y + G Uz + CG (24,+Butrus), 4,2, 3, 4, 2, B. Ve field} [by equ. O = { {G4, + 2 Yat Cz Uz + LCq 4, + B Q 4z + Grus/ 4,6,9, G ,T, Br & field} {(6+44) 4, + (6 + B(4) Ug+ (ezt GV) 43) 462, 63, CQ,*, Bre field] let at & C4 = Br, Cat BG Eße and CzfCqV=B3 and clearly, B., B2, B3 EF (since 4, 2, 63, 64,2, B, re field) {B, 4, + Buhat B3 43 ) B, B2, B3 & field] span} 4,42,43} hence, statement E is true best statement. ie span{ 4,4 43, 44} = spans 4, 46, 43}