Math 220 - March 26, 1999

The Vector Space of Polynomials of Given Degree

• For each integer n >= 0 P_{n} denotes the vector space of all polynomials of degree n, i.e., all functions of the form

  c_{0} + c_{1} t + c_{2} t^{2} + ... + c_{n} t^{n} ,

  where the c_{k} are constants.

• If f_{k} is the function f_{k}(t) = t^{k}, 0 <= k <= n, then the sequence f_{0}, f_{1}, ... f_{n} is a basis of P_{n}.

• The dimension of the vector space P_{n} is n + 1.

• Proposition. P_{n} is equal to the vector space of all solutions of the differential equation D^{n+1} f = 0.

Recess Schedule

This course will meet on Wednesday, March 31, but will not meet on Monday, April 5. There will be a quiz on April 7.

Assignment for Monday, March 29

1. Let phi be the linear map from P_{2} to R^{3} defined by phi(f) = (f(0), f'(0), f''(0)), and let psi be the linear map from R^{3} to P_{2} defined by (psi(a_{1}, a_{2}, a_{3}))(t) = a_{1} + a_{2} t + a_{3} t^{2}/2.

   1. Compute the composition phi \circ psi.

   2. Compute the composition psi \circ phi.

   3. Compute the composition (phi \circ D \circ psi)(a_{1}, a_{2}, a_{3}).

   4. Find a 3 \times 3 matrix M such that the linear map from R^{3} to R^{3} given by M is equal to phi \circ D \circ psi.

2. Let g_{k}(t) = t^{k}/k! for 0 <= k <= n. What does Taylor's Theorem (from second semester calculus) tell us about these functions in relation to the vector space P_{n} ?

3. If M is an m \times n matrix, then the row space of M is the set of all vectors in R^{n} that are linear combinations of the rows of M.

   1. Explain why the row space of M does not change when row operations are performed on M.

   2. Explain how to find a basis of the row space of M.

4. If M is an m \times n matrix, then the column space of M is the set of all vectors in R^{m} that are linear combinations of the columns of M.

   1. Use a simple example to show that the column space of M changes when row operations are performed on M.

   2. Explain why the answer to the question

      Which columns of M are linearly independent?

      does not change when row operations are performed on M.

   3. Explain how to find a basis of the column space of M.

   4. Explain why one always arrives at the same reduced row echelon form for a given matrix regardless of the particular sequence of row operations that is used to obtain it.

5. Explain why the row space of a matrix and the column space of a matrix must always have the same dimension.