Practice Problems on Vector Space Bases

A collection of exercises designed to reinforce your understanding of bases in vector spaces.

• Exercise 1
  Given two vectors in ℝ³, determine a basis that spans the space.
• Exercise 2
  Find both the basis and the dimension of a subspace W of ℝ³.
• Exercise 3
  Construct a basis for ℝ³ starting from two given vectors.
• Exercise 4
  Prove that ℝⁿ admits a basis consisting of exactly n linearly independent vectors.
• Exercise 5
  Determine a basis for a subspace of ℝ³ defined by a set of conditions.