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Exercise 2.3.3

Avg difficulty 3.0

Use the properties of scalar multiplication from Proposition 2.17 and the properties of vector addition from Proposition 2.12 to prove the following equalities. Justify every step.

  1. (k+ℓ)(u+v)=ku+kv+ℓu+ℓv(k+\ell)(\mathbf{u}+\mathbf{v})=k\mathbf{u}+k\mathbf{v}+\ell\mathbf{u}+\ell\mathbf{v}.

  2. 0u=00\mathbf{u}=\mathbf{0}.

  3. (−1)u=−u(-1)\mathbf{u}=-\mathbf{u}.

  4. −(ku)=k(−u)=(−k)u-(k\mathbf{u})=k(-\mathbf{u})=(-k)\mathbf{u}.

Selinger, Matrix Theory and Linear Algebra, Vectors in Rn, Scalar multiplication, Exercise 2.3.3

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Adapted from Matrix Theory and Linear Algebra © 2017–2026 Peter Selinger, Lyryx Learning, and Ken Kuttler. Licensed under CC BY 4.0. Source: Matrix Theory and Linear Algebra. Imported at revision d74bc3a.