Lab 22: Duality and Cohomology

Lab goals

This lab helps you practice the computational side of Chapter 22.

You will learn to:

compute dual basis functionals from an inverse matrix;
interpret rows of a matrix as linear measurements;
compute annihilators using null spaces of transposes;
build boundary and coboundary matrices;
compare homology and cohomology for hollow and filled triangles.

Python practice notebook

Download and complete the independent-study notebook:

The notebook includes worked solutions and similar practice questions.

Interactive lab

Open the interactive page:

Open Lab 22 interactive HTML

The interactive lab lets you experiment with:

dual basis extraction;
annihilators and fundamental subspaces;
hollow versus filled triangle cohomology.

Quick review

Dual basis

\[ P=[v_1\ \cdots\ v_n], \]

then the rows of $P^{-1}$ are the dual basis functionals.

Dual map

If $T$ has matrix $A$, then the dual map $T^*$ is represented by

\[ A^T. \]

Annihilator

For a subspace $U\le V$,

\[ U^\circ=\{\ell\in V^*:\ell(u)=0\text{ for all }u\in U\}. \]

If $V$ is finite-dimensional, then

\[ \dim U^\circ=\dim V-\dim U. \]

Cohomology

If $D_p$ is the matrix of $\partial_p$, then the corresponding coboundary matrix is

\[ D_p^T. \]

For finite-dimensional complexes over a field,

\[ \dim H^p=\dim H_p. \]

Independent-study questions with solutions

Question 1

Let

\[ v_1=(1,1),\qquad v_2=(1,2). \]

Find the dual basis.

Show solution

Form

\[ P=\begin{bmatrix}1&1\\1&2\end{bmatrix}. \]

Then

\[ P^{-1}=\begin{bmatrix}2&-1\\-1&1\end{bmatrix}. \]

Thus

\[ \varphi^1(x,y)=2x-y, \qquad \varphi^2(x,y)=-x+y. \]

Similar practice 1

Let

\[ v_1=(1,2),\qquad v_2=(3,5). \]

Find the dual basis.

Show solution

Here

\[ P=\begin{bmatrix}1&3\\2&5\end{bmatrix}, \qquad P^{-1}= \begin{bmatrix} -5&3\\ 2&-1 \end{bmatrix}. \]

\[ \varphi^1(x,y)=-5x+3y, \qquad \varphi^2(x,y)=2x-y. \]

Question 2

Let

\[ U=\operatorname{span}\{(1,1,0),(0,1,1)\}\subseteq\mathbb R^3. \]

Find one nonzero element of $U^\circ$.

Show solution

Let $\ell(x,y,z)=ax+by+cz$. We need

\[ \ell(1,1,0)=a+b=0, \qquad \ell(0,1,1)=b+c=0. \]

Choose $b=1$. Then $a=-1$ and $c=-1$. Hence

\[ \ell(x,y,z)=-x+y-z. \]

Similar practice 2

Let

\[ U=\operatorname{span}\{(1,0,1),(1,1,0)\}\subseteq\mathbb R^3. \]

Find one nonzero element of $U^\circ$.

Show solution

Let $\ell(x,y,z)=ax+by+cz$. We need

\[ a+c=0,\qquad a+b=0. \]

Choose $a=1$. Then $b=-1$ and $c=-1$. Hence

\[ \ell(x,y,z)=x-y-z. \]

Question 3

For the hollow triangle with three vertices and three edges, compute $\beta_0$ and $\beta_1$.

Show solution

The boundary matrix $D_1$ has rank $2$. Since there are no $2$-simplices, $D_2$ has rank $0$.

Thus

\[ \beta_0=n_0-\operatorname{rank}D_1=3-2=1, \]

and

\[ \beta_1=n_1-\operatorname{rank}D_1-\operatorname{rank}D_2=3-2-0=1. \]

Similar practice 3

For the filled triangle, compute $\beta_0$ and $\beta_1$.

Show solution

Now $\operatorname{rank}D_1=2$ and $\operatorname{rank}D_2=1$. Therefore

\[ \beta_0=3-2=1, \]

and

\[ \beta_1=3-2-1=0. \]

Reflection questions

Why are rows naturally interpreted as linear functionals?
Why does the dual map reverse the direction of a linear map?
Why do coboundary matrices appear as transposes of boundary matrices?
Why does adding a filled triangle kill the one-dimensional first cohomology class?
In what sense is a cohomology class a measurement of a hole?

# Lab 22: Duality and Cohomology {.unnumbered} # Lab goals This lab helps you practice the computational side of Chapter 22. You will learn to: - compute dual basis functionals from an inverse matrix; - interpret rows of a matrix as linear measurements; - compute annihilators using null spaces of transposes; - build boundary and coboundary matrices; - compare homology and cohomology for hollow and filled triangles. # Python practice notebook Download and complete the independent-study notebook: - [Download Lab 22 Python notebook](lab-22-duality-cohomology-independent-study.ipynb) - [Open Lab 14 in Google Colab](https://colab.research.google.com/github/wanghemath/MATH5110/blob/main/Labs/lab-22-duality-cohomology-independent-study.ipynb) The notebook includes worked solutions and similar practice questions. # Interactive lab Open the interactive page: [Open Lab 22 interactive HTML](lab-22-interactive.html) The interactive lab lets you experiment with: - dual basis extraction; - annihilators and fundamental subspaces; - hollow versus filled triangle cohomology. # Quick review ## Dual basis If $$ P=[v_1\ \cdots\ v_n], $$ then the rows of $P^{-1}$ are the dual basis functionals. ## Dual map If $T$ has matrix $A$, then the dual map $T^*$ is represented by $$ A^T. $$ ## Annihilator For a subspace $U\le V$, $$ U^\circ=\{\ell\in V^*:\ell(u)=0\text{ for all }u\in U\}. $$ If $V$ is finite-dimensional, then $$ \dim U^\circ=\dim V-\dim U. $$ ## Cohomology If $D_p$ is the matrix of $\partial_p$, then the corresponding coboundary matrix is $$ D_p^T. $$ For finite-dimensional complexes over a field, $$ \dim H^p=\dim H_p. $$ # Independent-study questions with solutions ## Question 1 Let $$ v_1=(1,1),\qquad v_2=(1,2). $$ Find the dual basis. <details> <summary>Show solution</summary> Form $$ P=\begin{bmatrix}1&1\\1&2\end{bmatrix}. $$ Then $$ P^{-1}=\begin{bmatrix}2&-1\\-1&1\end{bmatrix}. $$ Thus $$ \varphi^1(x,y)=2x-y, \qquad \varphi^2(x,y)=-x+y. $$ </details> ## Similar practice 1 Let $$ v_1=(1,2),\qquad v_2=(3,5). $$ Find the dual basis. <details> <summary>Show solution</summary> Here $$ P=\begin{bmatrix}1&3\\2&5\end{bmatrix}, \qquad P^{-1}= \begin{bmatrix} -5&3\\ 2&-1 \end{bmatrix}. $$ So $$ \varphi^1(x,y)=-5x+3y, \qquad \varphi^2(x,y)=2x-y. $$ </details> ## Question 2 Let $$ U=\operatorname{span}\{(1,1,0),(0,1,1)\}\subseteq\mathbb R^3. $$ Find one nonzero element of $U^\circ$. <details> <summary>Show solution</summary> Let $\ell(x,y,z)=ax+by+cz$. We need $$ \ell(1,1,0)=a+b=0, \qquad \ell(0,1,1)=b+c=0. $$ Choose $b=1$. Then $a=-1$ and $c=-1$. Hence $$ \ell(x,y,z)=-x+y-z. $$ </details> ## Similar practice 2 Let $$ U=\operatorname{span}\{(1,0,1),(1,1,0)\}\subseteq\mathbb R^3. $$ Find one nonzero element of $U^\circ$. <details> <summary>Show solution</summary> Let $\ell(x,y,z)=ax+by+cz$. We need $$ a+c=0,\qquad a+b=0. $$ Choose $a=1$. Then $b=-1$ and $c=-1$. Hence $$ \ell(x,y,z)=x-y-z. $$ </details> ## Question 3 For the hollow triangle with three vertices and three edges, compute $\beta_0$ and $\beta_1$. <details> <summary>Show solution</summary> The boundary matrix $D_1$ has rank $2$. Since there are no $2$-simplices, $D_2$ has rank $0$. Thus $$ \beta_0=n_0-\operatorname{rank}D_1=3-2=1, $$ and $$ \beta_1=n_1-\operatorname{rank}D_1-\operatorname{rank}D_2=3-2-0=1. $$ </details> ## Similar practice 3 For the filled triangle, compute $\beta_0$ and $\beta_1$. <details> <summary>Show solution</summary> Now $\operatorname{rank}D_1=2$ and $\operatorname{rank}D_2=1$. Therefore $$ \beta_0=3-2=1, $$ and $$ \beta_1=3-2-1=0. $$ </details> # Reflection questions 1. Why are rows naturally interpreted as linear functionals? 2. Why does the dual map reverse the direction of a linear map? 3. Why do coboundary matrices appear as transposes of boundary matrices? 4. Why does adding a filled triangle kill the one-dimensional first cohomology class? 5. In what sense is a cohomology class a measurement of a hole?