Range Space and Null Space

Let $h\colon\mathcal{P}_{3}\to\mathcal{P}_{4}$ be given by $p(x)\mapsto x\cdot p(x)$. Which of these are in the null space? Which are in the range space?

• (a)  $x^{3}$     (b)  $0$     (c)  $7$     (d)  $12x-0.5x^{3}$     (e)  $1+3x^{2}-x^{3}$

Find the range space and the rank of each homomorphism.

• $h\colon\mathcal{P}_{3}\to\mathbb{R}^{2}$ given by

• $f\colon\mathbb{R}^{2}\to\mathbb{R}^{3}$ given by

Find the range space and rank of each map.

• $h\colon\mathbb{R}^{2}\to\mathcal{P}_{3}$ given by

• $h\colon\mathcal{M}_{2\!\times\!2}\to\mathbb{R}$ given by

• $h\colon\mathcal{M}_{2\!\times\!2}\to\mathcal{P}_{2}$ given by

• the zero map $Z\colon\mathbb{R}^{3}\to\mathbb{R}^{4}$

For each linear map in the prior exercise, find the null space and nullity.

Find the nullity of each map below.

• (a)  $h\colon\mathbb{R}^{5}\to\mathbb{R}^{8}$ of rank five     (b)  $h\colon\mathcal{P}_{3}\to\mathcal{P}_{3}$ of rank one     (c)  $h\colon\mathbb{R}^{6}\to\mathbb{R}^{3}$, an onto map     (d)  $h\colon\mathcal{M}_{3\!\times\!3}\to\mathcal{M}_{3\!\times\!3}$, onto

What is the null space of the differentiation transformation $d/dx\colon\mathcal{P}_{n}\to\mathcal{P}_{n}$? What is the null space of the second derivative, as a transformation of $\mathcal{P}_{n}$? The $k$-th derivative?

For the map $h\colon\mathbb{R}^{3}\to\mathbb{R}^{2}$ given by

find the range space, rank, null space, and nullity.

Example 2.7 restates the first condition in the definition of homomorphism as ‘the shadow of a sum is the sum of the shadows’. Restate the second condition in the same style.

For the homomorphism $h\colon\mathcal{P}_{3}\to\mathcal{P}_{3}$ given by $h(a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3})=a_{0}+(a_{0}+a_{1})x+(a_{2}+a_{3})x^{3}$ find these.

• (a)  $\mathscr{N}(h)$     (b)  $h^{-1}(2-x^{3})$     (c)  $h^{-1}(1+x^{2})$

Each of these transformations of $\mathcal{P}_{3}$ is one-to-one. For each, find the inverse.

• $a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}\mapsto a_{0}+a_{1}x+2a_{2}x^{2}+3a_{3}x^{3}$

• $a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}\mapsto a_{0}+a_{2}x+a_{1}x^{2}+a_{3}x^{3}$

• $a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}\mapsto a_{1}+a_{2}x+a_{3}x^{2}+a_{0}x^{3}$

• $a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}\mapsto a_{0}+(a_{0}+a_{1})x+(a_{0}+a_{1}+a_{2})x^{2}+(a_{0}+a_{1}+a_{2}+a_{3})x^{3}$

Describe the null space and range space of a transformation given by $\vec{v}\mapsto 2\vec{v}$.

List all pairs $(\text{rank}(h),\text{nullity}(h))$ that are possible for linear maps from $\mathbb{R}^{5}$ to $\mathbb{R}^{3}$.

Does the differentiation map $d/dx\colon\mathcal{P}_{n}\to\mathcal{P}_{n}$ have an inverse?

Find the nullity of this map $h\colon\mathcal{P}_{n}\to\mathbb{R}$.

• Prove that a homomorphism is onto if and only if its rank equals the dimension of its codomain.

• Conclude that a homomorphism between vector spaces with the same dimension is one-to-one if and only if it is onto.

Show that a linear map is one-to-one if and only if it preserves linear independence.

Corollary 2.17 says that for there to be an onto homomorphism from a vector space $V$ to a vector space $W$, it is necessary that the dimension of $W$ be less than or equal to the dimension of $V$. Prove that this condition is also sufficient; use Theorem 1.9 to show that if the dimension of $W$ is less than or equal to the dimension of $V$, then there is a homomorphism from $V$ to $W$ that is onto.

Recall that the null space is a subset of the domain and the range space is a subset of the codomain. Are they necessarily distinct? Is there a homomorphism that has a nontrivial intersection of its null space and its range space?

Prove that the image of a span equals the span of the images. That is, where $h\colon V\to W$ is linear, prove that if $S$ is a subset of $V$ then $h([S])$ equals $[h(S)]$. This generalizes Lemma 2.1 since it shows that if $U$ is any subspace of $V$ then its image $\{\hskip 1.00006pth(\vec{u})\mid\vec{u}\in U\hskip 1.00006pt\}$ is a subspace of $W$, because the span of the set $U$ is $U$.

• Prove that for any linear map $h\colon V\to W$ and any $\vec{w}\in W$, the set $h^{-1}(\vec{w})$ has the form

  (if $h$ is not onto and $\vec{w}$ is not in the range of $h$ then this set is empty since its third condition cannot be satisfied). Such a set is a coset of $\mathscr{N}(h)$ and we denote it as $\vec{v}+\mathscr{N}(h)$.

• Consider the map $t\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ given by

  for some scalars $a$, $b$, $c$, and $d$. Prove that $t$ is linear.

• Conclude from the prior two items that for any linear system of the for…

Prove that for any transformation $t\colon V\to V$ that is rank one, the map given by composing the operator with itself $t\mathbin{\circ}t\colon V\to V$ satisfies $t\mathbin{\circ}t=r\cdot t$ for some real number $r$.

Let $h\colon V\to\mathbb{R}$ be a homomorphism, but not the zero homomorphism. Prove that if $\langle\vec{\beta}_{1},\dots,\vec{\beta}_{n}\rangle$ is a basis for the null space and if $\vec{v}\in V$ is not in the null space then $\langle\vec{v},\vec{\beta}_{1},\dots,\vec{\beta}_{n}\rangle$ is a basis for the entire domain $V$.

Show that for any space $V$ of dimension $n$, the dual space

is isomorphic to $\mathbb{R}^{n}$. It is often denoted $V^{\ast}$. Conclude that $V^{\ast}\cong V$.

Show that any linear map is the sum of maps of rank one.

Is ‘is homomorphic to’ an equivalence relation? (Hint: the difficulty is to decide on an appropriate meaning for the quoted phrase.)

Show that the range spaces and null spaces of powers of linear maps $t\colon V\to V$ form descending

and ascending

chains. Also show that if $k$ is such that $\mathscr{R}(t^{k})=\mathscr{R}(t^{k+1})$ then all following range spaces are equal: $\mathscr{R}(t^{k})=\mathscr{R}(t^{k+1})=\mathscr{R}(t^{k+2})\ldots\,$. Similarly, if $\mathscr{N}(t^{k})=\mathscr{N}(t^{k+1})$ then $\mathscr{N}(t^{k})=\mathscr{N}(t^{k+1})=\mathscr{N}(t^{k+2})=\ldots\,$.