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

Q1:

Let w∈ℝ and 𝑀==⟨𝑢,𝑢,𝑢,𝑢⟩∈ℝ∶⋅=0uwu. Is 𝑀 a subspace of ℝ?

Q2:

Let 𝑀==⟨𝑢,𝑢,𝑢,𝑢⟩∈ℝ𝑢=𝑢=0u:. Is 𝑀 a subspace of ℝ?

Q3:

Let 𝑀==(𝑢,𝑢,𝑢,𝑢)∈ℝ∶𝑢≥0𝑖=1,2,3,4uforeach. Is 𝑀 a subspace of ℝ?

Q4:

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

Q5:

Let w and w be the given vectors in ℝ and define 𝑀==⟨𝑢,𝑢,𝑢,𝑢⟩∈ℝ∶⋅=0⋅=0uwuwuand. Is 𝑀 a subspace of ℝ?

Q6:

Let 𝑀==⟨𝑢,𝑢,𝑢,𝑢⟩∈ℝ∶𝑢≥𝑢u. Is 𝑀 a subspace of ℝ?

Q7:

Let 𝑀==⟨𝑢,𝑢,𝑢,𝑢⟩∈ℝ∶|𝑢|≤4u. Is 𝑀 a subspace of ℝ?

Q8:

Let 𝑀==⟨𝑢,𝑢,𝑢,𝑢⟩∈ℝ∶(𝑢)=1usin. Is 𝑀 a subspace of ℝ?

Q9:

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

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