Worksheet: Vector Spaces

In this worksheet, we will practice defining vector spaces and giving examples to them and geometrically visualizing vector subspaces in R² and R³.


Let w and 𝑀==𝑢,𝑢,𝑢,𝑢=0uwu. Is 𝑀 a subspace of ?

  • Ano
  • Byes


Let 𝑀==𝑢,𝑢,𝑢,𝑢𝑢=𝑢=0u:. Is 𝑀 a subspace of ?

  • Ano
  • Byes


Let 𝑀==(𝑢,𝑢,𝑢,𝑢)𝑢0𝑖=1,2,3,4uforeach. Is 𝑀 a subspace of ?

  • ANo
  • BYes


Fill in the blank: The union of the coordinate axes in is not a vector space over because .

  • Ait is not closed under multiplication
  • Bit is infinite
  • Cit is not closed under addition
  • Dit does not contain a zero vector


Let w and w be the given vectors in and define 𝑀==𝑢,𝑢,𝑢,𝑢=0=0uwuwuand. Is 𝑀 a subspace of ?

  • AYes
  • BNo


Let 𝑀==𝑢,𝑢,𝑢,𝑢𝑢𝑢u. Is 𝑀 a subspace of ?

  • Ano
  • Byes


Let 𝑀==𝑢,𝑢,𝑢,𝑢|𝑢|4u. Is 𝑀 a subspace of ?

  • Ano
  • Byes


Let 𝑀==𝑢,𝑢,𝑢,𝑢(𝑢)=1usin. Is 𝑀 a subspace of ?

  • Ayes
  • Bno


Consider 𝐶, which is the vector space of functions that are continuous on the interval [0,1]. Let 𝑆=𝑓𝐶𝑓(𝑥)𝑥=0.d Is 𝑆 a linear subspace of 𝐶?

  • ANo
  • BYes

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