Math 206 Graded Problems/Proofs #19 Due Thursday April 18

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Note: Both of these problems can be done in more than one way. Some ways are easier than others. If you take advantage of theorems, not just definitions, the problems will be easier.

1. Let {v₁, v₂, v₃} be a basis for a vector space V.
   Let u₁ = v₁, u₂= v₁+v₂, u₃= v₁+v₂+v₃.
   Prove that {u₁, u₂, u₃} is a basis for V.
   
   
   
2. In P₃, let p₁, p₂, p₃, p₄ be the functions defined by
   • p₁(x) = 1+x
   • p₂(x) = 2x²+2x³
   • p₃(x) = 3+3x³
   • p₄(x) = 4x+4x²+4x³.

   Prove that {p₁, p₂, p₃, p₄} is a basis for P₃.

• Alexia Sontag, Mathematics
• Wellesley College
• Date Created: January 4, 2001
• Last Modified: April 8, 2002
• Expires: June 30, 2001