Question: vector spaces for academics

Exercise Prove that (-1)u = - u in any vector space. Note that (-1)u means the number -1 is multiplied to the vector u, and - u means the negative vector in the fourth property of the definition of vector spaces.

Answer

Exercise Prove that (a1u1 + a2u2) + (b1u1 + b2u2) = (a1 + b1)u1 + (a2 + b2)u2 in any vector space.

Answer

Exercise Give a detailed reason why, in any vector space,

  • u + v = 0 ⇒ u = - v.

  • 3u + 2v - 4w = 0 ⇒ v = - 3/2 u + 2w.

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