Maths paper 5 Gunshot Questions -10M
Maths paper 5 *Gunshot* *Questions -10M* ***1. Let V(F) be a vector space. A non empty set WV. Prove that the necessary and sufficient condition for W to be a subspace of V is a, b€F and α,β€W⇒aα+bβ€ W.*** Or Let V(F) be a vector space and Let W ⊆ V The necessary and sufficient conditions for W to be a subspacer of V are i α€W,β€W⇒α-β€W, ii.α€W,α€F⇒aα€W? Or b) If W, and W, are two subspaces of a vector space V(F) then prove that W+W, is a subspace of Vand W+W=<W, W₂> 2. If W, and W, are two subspaces of a finite dimensional vector space V(F) then prove that dim (W1+W2)= dim W1+dim2 W-dim (W1∩W2)? Or *****If W is a subspaces of a finite dimensional vector space V(F) then prove that dim (V/W)= dim V-dim W?***** 3.Rank Nullity Theorem? 4.Cayley-Hamilton Theorem? 5.Bessels Inequality? Or 6.Schwartz's Inequality?