10. Prove that W₁ {(a₁, a2,..., an) € F": a₁ + a₂ + + an = 0} is a subspace of Fr, but W₂ = {(a1, a2,...,an) € F: a₁ + a₂ ++an = 1} is not. =

6. Let V be a finite-dimensional vector space and E a subspace. Prove that V = EE¹.

19 Prove or give counterexample: If V₁, V₂, U are subspaces of V such that V₁ + U = V₂ + U, then V₁ = V₂.

Define the subset V, of R3 to be the set of all vectors y in R satisfying x +y+z = r, where r is a real number. Determine all values of r for which V, is a subspace of R³. Justify your answer.

Suppose that U and V are subspaces of IR". Fill in the blank so that the following statements are true. If {u, v, w} is a linearly independent subset of U, then dim(U) If {a,b,c} is a spanning set for V, then dim(V) 3. 3.