Question

Problem #5: [3 marks] Let u and y be vectors in R. Consider the following statements. (i) u vl = ||0|| + ||v|| (ii) ||u + v||2 = ||u||2 + ||v||2 + 2(u'v) (iii) If au + bv = cu + dv and a, b, c, and d are all nonzero then u = 0 and v=0. Determine which of the above statements are True (1) or False (2). So, for example, if you think that the answers, in the above order, are True False,False, then you would enter '1,2,2' into the answer box below (without the quotes). Problem #5:

Answer 1

vectors ☺ luovi e lullt llvll is not troue for ur in IR? Taxe u= (2,0,0) E1R3 and v= (4,0,0) EIR? Then juvl = 1 (20,0), (4,0) = 181=8 and 11 ull+ |1v1l = √ 4+0+0 + √16toto = 2+426 we know that 846. This luovi ellullt llvll is not troue for rectors ubv in IR?

(19) lluvii? = 1148 + 111 + 2 (UV) This is true for all u, v in IR? pfz Ilutuli? = ((atu) o (uty)) = (u.u) + (u.v) + (v.u) + (vov) = 11u112+2 (UV) Allvie (lor). OOO (111) of autbr=eurdy and a, b, cand d are all non zero then u=0 and w=0, this statement is also not troue. Take u= (2,0,0) EIR and v= (4,00) EIR? let a=2, bat then autbu = 2(2,0,0) + 1(4,0,0) (8,0,0) Take e=8, b=-2, then cut du = 8(2,0,0) +(-2) (4-00) = 8,0,0)

Thus aufbra eurdy is satisfied and asb,c,d are all non zero but u and V is not necessarily the zero vector in TR² .. The required answer is [2,1,2]
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