A mapping is said to be onto (sometimes called surjective) if it's image is the entire codomain.

Is the linear map such that onto?

True or false: If is a linear mapping, and is a vector space, then is a subspace of .

Let f be a mapping such that where is the vector space of polynomials up to the term. (ie polynomials of the form )

Let f be defined such that

Is f a homomorphism?

The previous two problems showed how the dimension of the domain and codomain can be used to predict if it is possible for the mapping to be 1-to-1 or onto. Now we'll apply that knowledge to isomorphism.

Let f be a mapping such that . Also the vector space V has dimension 4 and the vector space W has dimension 8. What property of isomorphism can f NOT satisify.

A mapping is said to be onto (sometimes called surjective) if it's image is the entire codomain.

Is the linear map such that onto?
(Note this is called the zero mapping)

True or false: The identity mapping , is also considered a linear mapping, regardless of the vector space .

This problem deals with the zero map. I.e the map that takes all vectors to the zero vector.

Consider the mapping such that .

What is the the null space of ?

is the set of all polynomials of finite degree in .

Define a linear mapping as follows:

.

True or false: is a one-to-one and onto linear mapping.

In the previous question, we said an isomorphism cannot be between vector spaces of different dimension. But are all homomorphisms between vector spaces of the same dimension an isomorphism?

Consider the homomorphism . Is f an isomorphism?

Consider the mapping such that .

What is the the null space of ?