Math 220 - March 29, 1999

The Dimension Formula

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 Wednesday, March 31

  1. Let D be differentiation regarded as a linear map from P_{2} to P_{2}. Describe the kernel and the image of D as subspaces of P_{2}.

  2. Let P be the union of all of the P_{n}, i.e., the vector space of all polynomials. (P is not finite-dimensional, but note that each polynomial has finite degree.)

    1. Find a basis of P.

    2. Regarding differentiation as a linear map from P to P, describe the kernel and the image of differentiation as subspaces of P.

  3. Give a simple example to show that the union of two subspaces of a given vector space need not be a subspace.

  4. Show that if V and W are vector spaces of the same finite dimension, then a linear map f from V to W is surjective (or ``onto'') if and only if it is injective (or ``one-to-one'').

  5. Give an example of an injective linear map from the vector space P of all polynomials to itself that is not surjective.


AUTHOR  |  COMMENT