- Suppose we're given a vector u in rectangular coordinates u = (x,y). How do we convert it to polar coordinates (r,θ)?
- Suppose we have two vectors u, v expressed in polar coordinates u = (r,θ) and v = (s,φ). How do we add them (assuming we also want the sum in polar coordinates)?
- Suppose you have a vector x in the plane with components a1, a2, expressed in the basis u1, u2. How do you write this vector in the standard basis e1 = (1,0), e2 = (0,1)? Do not assume that u1 and u2 are orthonormal.
- Now suppose x = (x1,x2) is instead expressed with respect to the standard basis e1 = (1,0), e2 = (0,1) i.e., x = x1 e1 + x2 e2. How do we re-write x in the orthonormal basis u1, u2? I.e., if we want to write it as x = a1 u1 + a2 u2, how do we get the coefficients a1, a2?
- Suppose we have the same setup as in the previous question, but u1 and u2 are no longer orthonormal. How do we now re-write x in the basis u1, u2? I.e., how do we get the coefficients a1, a2? How does the answer to this question relate to the answer to the previous question?
- Finally, suppose x is expressed in a non-orthonormal basis u1, u2, and we want to re-write it in another non-orthonormal basis v1, v2. What should we do?
Don't Be Dense
A lot of the matrices we work with in computer graphics (and other fields, like vision and machine learning) have only a small number of nonzero entries. Consider for instance the identity matrix, which has 1's along the diagonal; all other entries are 0. Other common matrices (such as the graph Laplacian) will also be mostly 0's. For this reason there are two basic ways to store a matrix A:
- Dense—Store every single entry Aij, whether or not it's equal to zero.
Sparse—Store only the entries where Aij does not equal zero (and implicitly assume all other entries are zero).
- How might you encode dense matrices (i.e., what kind of data structure?).
- How might you encode sparse matrices (i.e., what kind of data structure?).
- What's the cost of storing an n x n identity matrix using the two encodings described above?