26.14. Coordinate Systems and Change of Basis

Try these first

1. 1.

   What is a basis in a vector space?

   Hint

   Think about what building blocks are to a structure.

2. 2.

   How do you express a vector in terms of a basis?

   Hint

   It involves multiplication by coefficients.

3. 3.

   What is required to represent a vector in a different basis?

   • A different set of coordinates
   • A transformation matrix
   • A new vector space

   Hint

   Remember: responses depend on how bases relate.

4. 4.

   True or False: Every vector can be uniquely represented by its coordinates in a chosen basis.

   • True
   • False

   Hint

   Think about the definition of basis in vector spaces.

5. 5.

   Given two bases B = {(1,0), (0,1)} and C = {(1,1), (1,-1)}, determine the transformation matrix P and apply it to vector v = (2, 3).

   Hint

   Work out P by solving the linear combinations for each basis.

6. 6.

   If you have a vector that translates between two local and global coordinate systems, calculate the coordinates of the vector in the new system when provided with a transformation matrix.

   Hint

   Focus on applying the matrix transformation as the first step.