Question

(a) Construct the incidence matrix $A$ for the disconnected digraph $D$ in the figure. (b) Verify that $\operatorname{dim} \operatorname{ker} A=3$, which is the same as the number of connected components, meaning the maximal connected subgraphs in $D$. (c) Can you assign an interpretation to your basis for ker $A$ ? (d) Try proving the general statement that $\operatorname{dim} \operatorname{ker} A$ equals the number of connected components in the digraph $D$.

    (a) Construct the incidence matrix $A$ for the disconnected digraph $D$ in the figure. (b) Verify that $\operatorname{dim} \operatorname{ker} A=3$, which is the same as the number of connected components, meaning the maximal connected subgraphs in $D$. (c) Can you assign an interpretation to your basis for ker $A$ ? (d) Try proving the general statement that $\operatorname{dim} \operatorname{ker} A$ equals the number of connected components in the digraph $D$.

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 2, Problem 12

Instant Answer

Step 1

- Assume the digraph $D$ has vertices $v_1, v_2, \ldots, v_n$ and edges $e_1, e_2, \ldots, e_m$. - The incidence matrix $A$ is an $n \times m$ matrix where $A_{ij} = -1$ if edge $e_j$ starts at vertex $v_i$, $A_{ij} = 1$ if edge $e_j$ ends at vertex $v_i$, and Show more…

Show all steps

Thanks for your feedback!

(a) Construct the incidence matrix $A$ for the disconnected digraph $D$ in the figure. (b) Verify that $\operatorname{dim} \operatorname{ker} A=3$, which is the same as the number of connected components, meaning the maximal connected subgraphs in $D$. (c) Can you assign an interpretation to your basis for ker $A$ ? (d) Try proving the general statement that $\operatorname{dim} \operatorname{ker} A$ equals the number of connected components in the digraph $D$.

Play audio

Feedback

Ace is your personal tutor. It breaks down any question with clear steps so you can learn.

Start Using Ace

Continue solving this problem

Create a free account to:

View full step-by-step solution
Ask follow-up questions with Ace AI
Save progress and study later

Please give Ace some feedback