26.6. Basis and Dimension

Try these first

1. 1.

   Define what a basis is in the context of vector spaces.

   Hint

   Consider how vectors relate to each other in terms of dependencies.

2. 2.

   What is the dimension of R⁵?

   Hint

   Think about how many independent directions are in the space.

3. 3.

   What is a basis of a vector space?

   • A set of vectors that are not independent.
   • A linearly independent set of vectors that spans the space.
   • Any set of vectors in the vector space.

   Hint

   Think about the criteria a set must meet to be considered a basis.

4. 4.

   Does the zero vector space have a basis?

   • True
   • False

   Hint

   Consider how many independent vectors are necessary for a basis.

5. 5.

   Given a vector space consisting of vectors in R³, determine if the set {(1,0,0), (0,1,0), (0,0,1), (1,1,1)} is a basis. Explain your reasoning.

   Hint

   Check if the last vector can be formed using coefficients from the first three.

6. 6.

   If a vector space V has dimension n, show that no more than n vectors can be linearly independent in V.

   Hint

   Consider how adding vectors beyond the dimension interacts with independence.