Math 206 Graded Problems/Proofs #15 Due Monday April 1

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1. Assume that {u₁, u₂, u₃ } is a linearly dependent set in a vector space V, and v is a vector belonging to V. Prove that the set {u₁, u₂, u₃, v} is also linearly dependent.
   • Try to give a proof that would readily generalize to a dependent set with more than three vectors in it.
   • Hint: Writing your proof will be easiest if you use the following characterization of dependence.

   
   A set {{v₁, v₂, ..., v_r} is linearly dependent iff there exist scalars k₁, k₂, ...,k_r such that

   • at least one of the numbers k₁, k₂, ...,k_r is different from 0 and
   • k₁v₁ +k₂v₂ +...+k_rv_r = 0.

   
   
   

2. Assume that {u₁, u₂, u₃ } is a linearly independent set in some vector space V, and that k₁, k₂, and k₃ are nonzero scalars. Prove that the set {k₁u₁, k₂ u₂, k₃ u₃ }is linearly independent. For your proof, use the following outline. (So I'm insisting on a particular plan of attack this time.There are other plans that will work. You should feel free to try them. But you must use this plan for the proof that you want to have graded.)
   • Assume that c₁, c₂, c₃ are real numbers for which c₁(k₁u₁) + c₂(k₂u₂) +c₃(k₃u₃) = 0.
   • Provide reasoning to show that c₁ = 0, c₂ = 0, c₃ = 0.

    

• Alexia Sontag, Mathematics
• Wellesley College
• Date Created: January 4, 2001
• Last Modified: March 24, 2002
• Expires: June 30, 2002