24.8. Coordinates of a Vector

Try these first

1. 1.

   If vector v = (5, 2) is expressed in terms of basis B = {(1, 0), (0, 1)}, what is the coordinate vector?

   Hint

   Identify the coefficients based on the basis representation.

2. 2.

   Express vector v = (3, 3) in terms of basis B = {(1, 1), (1, 0)}. What is the coordinate vector?

   Hint

   Think about how many times you need each basis vector.

3. 3.

   What is a coordinate vector?

   • A representation of a vector in a visual form
   • The tuple of coefficients expressing a vector in terms of a basis
   • A way to graph vectors

   Hint

   Think about representing vectors in terms of their basis.

4. 4.

   True or False: All vectors in a vector space can be expressed in terms of the same basis.

   • True
   • False

   Hint

   Recall the properties of vector spaces and their bases.

5. 5.

   Given basis B = {(1, 0, 0), (0, 1, 0), (0, 0, 1)} and vector v = (5, 7, 2), compute and explain the coordinates of vector v.

   Hint

   Use the definition of coordinate vectors and linear combinations.

6. 6.

   If we alter the basis to B' = {(2, 0), (0, 3)} and have vector v = (6, 9), find the coordinates of v in this new basis.

   Hint

   Express the vector in terms of the new basis and solve for the coefficients.