Self-Composition

Give the chains of range spaces and null spaces for the zero and identity transformations.

For each map, give the chain of range spaces and the chain of null spaces, and the generalized range space and the generalized null space.

• $t_{0}\colon\mathcal{P}_{2}\to\mathcal{P}_{2}$, $a+bx+cx^{2}\mapsto b+cx^{2}$

• $t_{1}\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$,

• $t_{2}\colon\mathcal{P}_{2}\to\mathcal{P}_{2}$, $a+bx+cx^{2}\mapsto b+cx+ax^{2}$

• $t_{3}\colon\mathbb{R}^{3}\to\mathbb{R}^{3}$,

Prove that function composition is associative $(t\mathbin{\circ}t)\mathbin{\circ}t=t\mathbin{\circ}(t\mathbin{\circ}t)$ and so we can write $t^{3}$ without specifying a grouping.

Check that a subspace must be of dimension less than or equal to the dimension of its superspace. Check that if the subspace is proper (the subspace does not equal the superspace) then the dimension is strictly less. (This is used in the proof of Lemma 1.4.)

Prove that the generalized range space $\mathscr{R}_{\infty}(t)$ is the entire space, and the generalized null space $\mathscr{N}_{\infty}(t)$ is trivial, if the transformation $t$ is nonsingular. Is this ‘only if’ also?

Verify the null space half of Lemma 1.4.

Give an example of a transformation on a three dimensional space whose range has dimension two. What is its null space? Iterate your example until the range space and null space stabilize.

Show that the range space and null space of a linear transformation need not be disjoint. Are they ever disjoint?