Answer Happy

QUESTION 4 4.1 Which of the following sets of vectors are linearly dependent? 3 4.1.1 C₁ = ₁-(-)-(1) and c₂ = 2 4 3 4.1.2 C₁ = (-1) ₁²2 - (²) 2 and c3 = , C2 = 5 (63) 2 3 4 1 2 41.2 - - ( ) --- () ---- () = 2 3 = and c3 = 3 2 3 ^₁^4₁-(-). - ( ) - ( ) ( ) 4.1.4 = C3 = 3 and C4 = , C₂ = 3 5 4.2 Let S₁ be the vector space spanned by the vectors 3 2 1 --0---0----- () = and x3 = , X2 = 3 2 1 1 1 whereas the vector space S₂ is spanned by the vectors 3 1 0 2 -- ()--- () ---- () = = and y3 = 5 -1 1 Find the following: (3) 4.2.1 Bases for S₁ and $₂. 4.2.2 The dimension of S₁ + S₂. 4.2.3 A basis of S₁ + S₂. (4) (3) STA3710/011/0/2022 4.2.4 The dimension of S₁ S₂. (3) (3) 4.2.5 A basis for S₁ S₂. 4.3 Let S₁ and S₂ be vector subspaces of R" with dim (S₁) = r₁ and dim (S₂) = 12. 4.3.1 Find expressions in terms of m, r₁ and r₂ for the smallest and largest possible values of dim (S₁ + S₂). (3) 4.3.2 Find the smallest and largest possible values of dim (S₁ S₂). (4) 4 [35 Marks] (2) (3) (3) (4)