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.

   Proof. From our assumption that {u₁, u₂, u₃ }is linearly dependent, we know (see the hint) that there exist scalars k₁, k₂, k₃ such that

   • at least one of the numbers k₁, k₂,k₃ is different from 0 and
   • k₁u₁ +k₂u₂ +k₃u₃ = 0.
     

   Set c₁= k₁, c₂= k₂, c₃=k₃ and c₄=0. Then (remember that 0w is always 0)
   

   • at least one of the numbers c₁, c₂,c₃, c₄ is different from 0 and
   • c₁u₁ +c₂u₂ +c₃u₃ +c₄v = 0.
     

   Using the same characterization of dependence as before, we conclude that {u₁, u₂, u₃, v} is dependent.
   
   Note: It is possible to do this proof using the other characterization of dependence (at least one vector in the purportedly dependent set is a linear combination of the others) but it's more awkward to write such a proof, since one has to deal, somehow, with not knowing which vector can be written in terms of the others. The key idea, however, remains the same: adding in 0w just adds in 0, hence preserves/extends the dependency relation that already exists.
   
   

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.

   
   Proof (following the suggested/required outline).

   • Assume that c₁, c₂, c₃ are real numbers for which
     c₁(k₁u₁) + c₂(k₂u₂) +c₃(k₃u₃) = 0.
     

     Vector-space axioms allow us to rewrite this as
     

     (c₁k₁)u₁) + (c₂k₂)u₂) +(c₃k₃)u₃= 0.
     

     Since {u₁, u₂, u₃ } is linearly independent we must have

     c₁k₁=0
     
     c₂k₂=0
     
     c₃k₃=0.
     

     Since k₁, k₂, and k₃ all differ from zero, a property of real numbers gives us that
     

   • c₁=c₂=c₃=0

     as required. This completes the proof.

• Alexia Sontag, Mathematics
• Wellesley College
• Date Created: January 4, 2001
• Last Modified: April 17, 2002
• Expires: June 30, 2002