Question

1 3. Consider the vector v= (-1) in R3. Let U = {w € R3 :w.v=0}, where w.v is the dot product. 2 (a) Prove that U is a subspace of R3. (b) Find a basis for U and compute its dimension. 4. Decide whether or not the following subsets of vector spaces are linearly independent. If they are, prove it. If they aren't, write one as a linear combination of the others. (a) The subset {0 0 0 of R3 (b) The subset (1 + x – 2x2,2 – 3 + x2,1 - 2x + 3x2} of P2(R).

Answer 1

3/ Givem, U = {WER? : W.V = 0] 2= ( 1 ) let a BEU, a, bt IR i div=0 and B. r=0. ( h adtbl a now, consider ad+ b3 which belags to R² as (ad+bB). V = a(d.v) + b (B.V) ß belongs to 3 ( = a.otb.o dot product [by (17] is distral sective 2020-4-10 C d, ßEU - adtbße u

U is a subspace of R². (6) let w- f er? such kow, that worth = 0 X-+2200 - 0 - y = x+22 -M4,00) 741,00)+7(200) 11,00 + 17, -1,0) (272) = (a + 22, 2) - *1,,0) + 210.2.1) .. () = ()+2 () atso artt, t0) + bt also a. l ) + b ( 2 ) = 0 > a=0cb. {(5).(3) LII- and spans u {1:0)) a basis for L'Io and spans u and dimension of this basis is 2. 41 (a) For (a) For the let given subset of IR a / 11 Let a:(+48)+(!)-o Wancer va, L. CER - act 3b + c =0 20+21 = 0 3a +6-20=0 - (1) - lül lui)

consider the matrix system of the above ea's -2 -1-2 1 3 It 2 11. = -1 +2 -1 =0 1 - 1-2 1-2 + 3 T The given subset {1+2-22h, 2 -x + x², 1-2x+327 of P2 (R) is lineaply dependent. how, 2-x+ x = (1 +2 - 2x + (1-2x +30) It is peppesentation of one vector as a linear combination of others.