24.6. Basis

Try these first

1. 1.

   What is a basis in a vector space?

   Hint

   Think about the minimal number of vectors needed to describe all vectors in the space.

2. 2.

   Give an example of a basis for R².

   Hint

   Consider the axes in a two-dimensional space.

3. 3.

   What is a basis of a vector space?

   • A single vector
   • A set of vectors
   • A linearly dependent set

   Hint

   Remember the properties that define a basis.

4. 4.

   True or False: The vectors (1,2) and (1,2) can form a basis for R².

   • True
   • False

   Hint

   Consider if the vectors can be combined to give you the identity vector.

5. 5.

   Find a basis for the vector space spanned by the vectors {(2,4), (-1,-2), (3,6)}.

   Hint

   Identify which vectors are linearly dependent.

6. 6.

   Determine if the set {(1,1), (2,2), (3,4)} is a basis for R². Explain your reasoning.

   Hint

   Try expressing one vector as a combination of others.