Definition

Decide if each $h\colon\mathbb{R}^{3}\to\mathbb{R}^{2}$ is linear.

• (a)  $h(\begin{pmatrix}x\\ y\\ z\end{pmatrix})=\begin{pmatrix}x\\ x+y+z\end{pmatrix}$     (b)  $h(\begin{pmatrix}x\\ y\\ z\end{pmatrix})=\begin{pmatrix}0\\ 0\end{pmatrix}$     (c)  $h(\begin{pmatrix}x\\ y\\ z\end{pmatrix})=\begin{pmatrix}1\\ 1\end{pmatrix}$     (d)  $h(\begin{pmatrix}x\\ y\\ z\end{pmatrix})=\begin{pmatrix}2x+y\\ 3y-4z\end{pmatrix}$

Decide if each map $h\colon\mathcal{M}_{2\!\times\!2}\to\mathbb{R}$ is linear.

• $h(\begin{pmatrix}a&b\\ c&d\end{pmatrix})=a+d$

• $h(\begin{pmatrix}a&b\\ c&d\end{pmatrix})=ad-bc$

• $h(\begin{pmatrix}a&b\\ c&d\end{pmatrix})=2a+3b+c-d$

• $h(\begin{pmatrix}a&b\\ c&d\end{pmatrix})=a^{2}+b^{2}$

Show that these are homomorphisms. Are they inverse to each other?

• $d/dx\colon\mathcal{P}_{3}\to\mathcal{P}_{2}$ given by $a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}$ maps to $a_{1}+2a_{2}x+3a_{3}x^{2}$

• $\int\colon\mathcal{P}_{2}\to\mathcal{P}_{3}$ given by $b_{0}+b_{1}x+b_{2}x^{2}$ maps to $b_{0}x+(b_{1}/2)x^{2}+(b_{2}/3)x^{3}$

Is (perpendicular) projection from $\mathbb{R}^{3}$ to the $xz$-plane a homomorphism? Projection to the $yz$-plane? To the $x$-axis? The $y$-axis? The $z$-axis? Projection to the origin?

Verify that each map is a homomorphism.

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

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

Show that, while the maps from Example 1.3 preserve linear operations, they are not isomorphisms.

Is an identity map a linear transformation?

Stating that a function is ‘linear’ is different than stating that its graph is a line.

• The function $f_{1}\colon\mathbb{R}\to\mathbb{R}$ given by $f_{1}(x)=2x-1$ has a graph that is a line. Show that it is not a linear function.

• The function $f_{2}\colon\mathbb{R}^{2}\to\mathbb{R}$ given by

  does not have a graph that is a line. Show that it is a linear function.

Part of the definition of a linear function is that it respects addition. Does a linear function respect subtraction?

Assume that $h$ is a linear transformation of $V$ and that $\langle\vec{\beta}_{1},\ldots,\vec{\beta}_{n}\rangle$ is a basis of $V$. Prove each statement.

• If $h(\vec{\beta}_{i})=\vec{0}$ for each basis vector then $h$ is the zero map.

• If $h(\vec{\beta}_{i})=\vec{\beta}_{i}$ for each basis vector then $h$ is the identity map.

• If there is a scalar $r$ such that $h(\vec{\beta}_{i})=r\cdot\vec{\beta}_{i}$ for each basis vector then $h(\vec{v})=r\cdot\vec{v}$ for all vectors in $V$.

Consider the vector space $\mathbb{R}^{+}$ where vector addition and scalar multiplication are not the ones inherited from $\mathbb{R}$ but rather are these: $a+b$ is the product of $a$ and $b$, and $r\cdot a$ is the $r$-th power of $a$. (This was shown to be a vector space in an earlier exercise.) Verify that the natural logarithm map $\ln\colon\mathbb{R}^{+}\to\mathbb{R}$ is a homomorphism between these two spaces. Is it an isomorphism?

Consider this transformation of the plane $\mathbb{R}^{2}$.

Find the image under this map of this ellipse.

Imagine a rope wound around the earth’s equator so that it fits snugly (suppose that the earth is a sphere). How much extra rope must we add so that around the entire world the rope will now be six feet off the ground?

Verify that this map $h\colon\mathbb{R}^{3}\to\mathbb{R}$

is linear. Generalize.

Show that every homomorphism from $\mathbb{R}^{1}$ to $\mathbb{R}^{1}$ acts via multiplication by a scalar. Conclude that every nontrivial linear transformation of $\mathbb{R}^{1}$ is an isomorphism. Is that true for transformations of $\mathbb{R}^{2}$? $\mathbb{R}^{n}$?

Show that for any scalars $a_{1,1},\dots,a_{m,n}$ this map $h\colon\mathbb{R}^{n}\to\mathbb{R}^{m}$ is a homomorphism.

Consider the space of polynomials $\mathcal{P}_{n}$.

• Show that for each $i$, the $i$-th derivative operator $d^{i}/dx^{i}$ is a linear transformation of that space.

• Conclude that for any scalars $c_{k},\ldots,c_{0}$ this map is a linear transformation of that space.

Lemma 1.17 shows that a sum of linear functions is linear and that a scalar multiple of a linear function is linear. Show also that a composition of linear functions is linear.

Where $f\colon V\to W$ is linear, suppose that $f(\vec{v}_{1})=\vec{w}_{1}$, …, $f(\vec{v}_{n})=\vec{w}_{n}$ for some vectors $\vec{w}_{1}$, …, $\vec{w}_{n}$ from $W$.

• If the set of $\vec{w}\,$’s is independent, must the set of $\vec{v}\,$’s also be independent?

• If the set of $\vec{v}\,$’s is independent, must the set of $\vec{w}\,$’s also be independent?

• If the set of $\vec{w}\,$’s spans $W$, must the set of $\vec{v}\,$’s span $V$?

• If the set of $\vec{v}\,$’s spans $V$, must the set of $\vec{w}\,$’s span $W$?

Generalize Example 1.16 by proving that for every appropriate domain and codomain the matrix transpose map is linear. What are the appropriate domains and codomains?

• Where $\vec{u},\vec{v}\in\mathbb{R}^{n}$, by definition the line segment connecting them is the set $\ell=\text{$\{\hskip 1.00006ptt\cdot\vec{u}+(1-t)\cdot\vec{v}\mid t\in[0..1]\hskip 1.00006pt\}$}$. Show that the image, under a homomorphism $h$, of the segment between $\vec{u}$ and $\vec{v}$ is the segment between $h(\vec{u})$ and $h(\vec{v})$.

• A subset of $\mathbb{R}^{n}$ is convex if, for any two points in that set, the line segment joining them lies entirely in that set. (The inside of a sphere is convex while the skin of a sphere is not.) Prove that linear maps from $\mathbb{R}^{n}$ t…

Let $h\colon\mathbb{R}^{n}\to\mathbb{R}^{m}$ be a homomorphism.

• Show that the image under $h$ of a line in $\mathbb{R}^{n}$ is a (possibly degenerate) line in $\mathbb{R}^{m}$.

• What happens to a $k$-dimensional linear surface?

Prove that the restriction of a homomorphism to a subspace of its domain is another homomorphism.

Assume that $h\colon V\to W$ is linear.

• Show that the range space of this map $\{\hskip 1.00006pth(\vec{v})\mid\vec{v}\in V\hskip 1.00006pt\}$ is a subspace of the codomain $W$.

• Show that the null space of this map $\{\hskip 1.00006pt\vec{v}\in V\mid h(\vec{v})=\vec{0}_{W}\hskip 1.00006pt\}$ is a subspace of the domain $V$.

• Show that if $U$ is a subspace of the domain $V$ then its image $\{\hskip 1.00006pth(\vec{u})\mid\vec{u}\in U\hskip 1.00006pt\}$ is a subspace of the codomain $W$. This generalizes the first item.

• Generalize the second item.

Consider the set of isomorphisms from a vector space to itself. Is this a subspace of the space $\mathop{{\cal L}}(V,V)$ of homomorphisms from the space to itself?

Does Theorem 1.9 need that $\langle\vec{\beta}_{1},\ldots,\vec{\beta}_{n}\rangle$ is a basis? That is, can we still get a well-defined and unique homomorphism if we drop either the condition that the set of $\vec{\beta}$’s be linearly independent, or the condition that it span the domain?

Let $V$ be a vector space and assume that the maps $f_{1},f_{2}\colon V\to\mathbb{R}^{1}$ are linear.

• Define a map $F\colon V\to\mathbb{R}^{2}$ whose component functions are the given linear ones.

  Show that $F$ is linear.

• Does the converse hold — is any linear map from $V$ to $\mathbb{R}^{2}$ made up of two linear component maps to $\mathbb{R}^{1}$?

• Generalize.