The definitions of image, fiber, and kernel from the handout of February 5 apply when one has a linear map from a vector space to a vector space. In particular, they apply even when the vector spaces are not finite-dimensional.
The definition of parallel subsets from the handout of March 12 applies in the context of general vector spaces. In particular, two subsets of a vector space are parallel if one is a translate of the other.
If f is a linear map from a vector space to a vector space, then any non-empty fiber of f is parallel to the kernel of f, and any two non-empty fibers of f are parallel to each other. Application: A linear map is injective if and only if the zero vector is the only vector in its kernel.
Theorem (The dimension formula.) If V and W are vector spaces with V finite dimensional and if f is a linear map from V to W, then both the image of f and the kernel of f are finite-dimensional, and one has
This course will meet on Wednesday, March 31, but will not meet on Monday, April 5. There will be a quiz on April 7.
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}.
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.)
Find a basis of P.
Regarding differentiation as a linear map from P to P, describe the kernel and the image of differentiation as subspaces of P.
Give a simple example to show that the union of two subspaces of a given vector space need not be a subspace.
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'').
Give an example of an injective linear map from the vector space P of all polynomials to itself that is not surjective.