Question

Question 3 (10 points) Let U be the subspace of C" defined by: U = {(21, 22, 23, 24, 25) € CÓ : 6z1 = z2 and 23 +224 + 3z5 = 0}. (a) Find a basis of U. (b) Extend the basis in part (a) to a basis of C".

*NEED HELP UNDERSTANDING THIS CONCEPT, NOT SURE IF MY ANSWER IS ON POINT*

*PLEASE SHOW ALL WORK AND STEP BY STEP SOLUTION*

Answer 1

(a) W = {(21, 22, 23, 24, 25) EOS : 62 22 and 23 +27 +32=0} Let & } = (a, b, c, d), e) ey. Then, a, b, c, d, e E es رما = ها tad +3 e o rais c = -2d-se Hence & = (a 69,- 22-3e, d, e) al16,0,0,0) + d(0, 0, 2, 1,0) telo,o – 3, 0, 1) let & = (1,6, 0, 0,0) = (0o,-2, 10). = (0,0,-3,0,1) Then, &ELSR, Br} Therefore, &EU & ELEK, B is. UCL$&, B, Y} - Again, & EU, BEU, VEO {& Brzeu From and we get U=L5L, B, V}. Hence the basis of u arest, B, de (6) let us examine if & = (0,0,0,1,0). belongs to Lf &, Br let & G & + CB at Cz? Where are the 9=0,69 = 0,26-36 = 0, C₂ = 1, G=0 This is an inconsistent system of egnations in 9,2,3. [ 9 = 1 and 420] Therefore, & 4 LET, B, & } . Consequently, the set food, Bd, 293 is a linearly independent in es where are

since as is a vector space of dinension 5, the set &&, B, D, E Containing tour vectors te can not be a basis of es. cet us cet is evenime de examine of ks = (1, 0, 0, 0, 0) 60, 650, belough to L & B, 0, 43. ne cet &s = dax derde B+ d38 + dara, a, e, desde El d =as, 6 d =0,- 2d2-3d3=0, dz+ da so ds=0 tant dystem it This is an ince I of dine meon equations in di, da, dz, da Therefore Es & L & B, V, Es} Consequently, the set &&, B, , , &} is a linearly independent set in @r. Eince 05 is a rector of space of dimension 5 and the set L, Breg, es} is a linearly independent set of 5 vectors ines (d, B, d, e, & } is a basis of a Clearly this basis set is an extension the given set of vectors {(1,6, 0, 0, 0), set (0,0, 2, 1,0), (0, -30, 1) Z