21.8.3. Basis and Dimension

Try these first

1. 1.

   What is a basis in vector space?

   Hint

   Think about how vectors can represent points in space.

2. 2.

   How is the dimension of a vector space determined?

   Hint

   Consider how many directions can be formed by the basis vectors.

3. 3.

   What is a basis in linear algebra?

   • A set of all vectors in a space
   • A set of linearly independent vectors that span the space
   • A vector that does not change direction

   Hint

   Remember the definition of linear combinations.

4. 4.

   Dimension refers to:

   • True
   • False

   Hint

   Think of how many directions the basis can point.

5. 5.

   Determine if the vectors (3,1,2), (2,0,0), and (0,4,1) are linearly independent. What is the dimension of their span?

   Hint

   Try arranging the vectors into a matrix to examine their independence.

6. 6.

   Given a vector space of dimension 5, what are possible bases for it, and how would you verify their independence?

   Hint

   Consider the geometric interpretation of vector directions.