Answer Happy

Do not assume anything more than what is written. 1. Let V be a finite dimensional vector space. Suppose that S is a subset of V which is linearly independent. Then |S| < dim(V). 2. Let V be a vector space. Let V₁, V2 E V. Suppose that there exists a vector v EV such that v V₁ V2 + v. Then v₁ = V2. - 3. {(a, 1): a ER} is a subspace of R². 4. Let V be a vector space over a field F. Let S = (v1, 02, 03} V. Suppose that S is linearly dependent and Oy S. Then there exists a1, a2 E F such that a₁v₁ + a₂. . V2 V3. 5. Let V and W be finite dimensional vector spaces. Let T: VW be a linear transformation. Then there exists a linear transformation U: W→ V such that U o T = IV. 6. Let V be a vector space and S be a non-empty subset of V. Then span (span (S)) = span(S). 7. Let V and W be vector spaces and T, U: VW be linear transformations. Then Z := {ve V: T(v) = U(v)} is a subspace of V. 8. Let S := {V1, V2, U3, U4, U5) C P2 (R) such that span (S) = P₂ (R). Then there exists a subset of S which is a basis for P₂ (R). 9. There exists a vector space V and a linear transformation T: VV such that Tis surjective but not injective. 10. Let V and W be finite dimensional vector spaces, B = {1,..., Un} be a basis for V, and T: V→ W be an injective linear transformation. Then {T(v₁),..., T(vn)} is a basis for W.