Theme. Real matrices come from applications and have structure. Strang calls the graph and its incidence matrix "the most important model in applied math." All four fundamental subspaces get a physical meaning here.
1. The graph and its incidence matrix
What a graph is. A graph consists of nodes and edges. Each edge is given a direction, drawn as an arrow. The direction is arbitrary; it only fixes which way counts as positive.
The example: \(n = 4\) nodes and \(m = 5\) edges.
| Edge | From → To |
|---|---|
| 1 | 1 → 2 |
| 2 | 2 → 3 |
| 3 | 1 → 3 |
| 4 | 1 → 4 |
| 5 | 3 → 4 |
The incidence matrix \(A\) (\(m \times n = 5 \times 4\)) has one row per edge and one column per node. Each row has \(-1\) at the node where the edge starts and \(+1\) at the node where it ends:
Structure:
- \(A\) is sparse: each row has exactly 2 nonzeros, so there are \(2m\) nonzeros in total.
- Loops give dependent rows. Edges 1, 2, 3 form a loop, and row 1 + row 2 = row 3.
Physical interpretation. Strang uses an electrical network: potentials, currents, Ohm's law. The same structure describes hydraulic networks (water or oil flow) and structures such as bridges or geodesic domes.
2. Nullspace \(N(A)\): constant potentials
What \(A\) computes. Let \(x = (x_1, x_2, x_3, x_4)\) be the potentials at the nodes. Then
When is \(Ax = 0\)? Exactly when every potential difference is zero, meaning all potentials are equal:
Physical meaning. Potentials are determined only up to an additive constant. Only differences make current flow. Strang compares this to the \(+C\) in an indefinite integral, and to temperature scales (Celsius vs. absolute zero).
Grounding a node. In practice, one node is fixed at potential 0, for example \(x_4 = 0\). That removes its column, and the remaining 3 columns are independent.
Rank:
3. Left nullspace \(N(A^\top)\): Kirchhoff's Current Law
Dimension. \(\dim N(A^\top) = m - r = 5 - 3 = 2\).
The transpose:
Meaning of \(y\). Let \(y = (y_1, \dots, y_5)\) be the currents on the edges.
Each row of \(A^\top y = 0\) is a balance equation at one node:
| Node | Equation | Meaning |
|---|---|---|
| 1 | \(-y_1 - y_3 - y_4 = 0\) | Edges 1, 3, 4 all leave node 1, so their currents sum to 0 |
| 2 | \(y_1 - y_2 = 0\) | Current in equals current out |
| 3 | \(y_2 + y_3 - y_5 = 0\) | Current in equals current out |
| 4 | \(y_4 + y_5 = 0\) | Current in equals current out |
This is Kirchhoff's Current Law (KCL): no charge accumulates at a node, so current in equals current out. Strang calls \(A^\top y = 0\) probably the most fundamental equation of applied mathematics.
Basis for \(N(A^\top)\): currents around loops
The basis can be read directly from the graph, without elimination. Any current that flows around a loop satisfies KCL.
Loop 1 → 2 → 3 → 1 (edges 1, 2, then back along edge 3, against its arrow):
Loop 1 → 3 → 4 → 1 (edges 3, 5, then back along edge 4, against its arrow):
The outer loop 1 → 2 → 3 → 4 → 1 gives \((1, 1, 0, -1, 1)\). This is not a new basis vector: it is the sum of the two above. The currents on edge 3 cancel.
\(\dim N(A^\top)\) = the number of independent loops.
4. Row space \(C(A^\top)\): trees
Dimension. \(\dim C(A^\top) = r = 3\). So 3 of the 5 rows of \(A\) (the edges) are independent.
Not just any three edges. Edges 1, 2, 3 are dependent, because they form a loop. The vector \((1,1,-1,0,0)\) in \(N(A^\top)\) records exactly this dependency.
Pivot columns of \(A^\top\). They are columns 1, 2, 4, which correspond to edges 1, 2, 4.
What those edges look like. Together they reach all 4 nodes using only 3 edges and contain no loop. Adding any other edge would create a loop.
A graph with no loops is a tree. Independent rows of \(A\) correspond to edges forming a tree. Dependencies among rows come from loops.
5. Euler's formula, from linear algebra
Combine three facts:
- \(\dim N(A^\top) = m - r\)
- \(\dim N(A^\top) = \) the number of independent loops
- \(r = n - 1\)
This gives
Rearranging:
This is Euler's formula. It holds for any connected graph. Strang notes the dimensions involved: nodes are 0-dimensional, edges 1-dimensional, and loops are 2-dimensional (they enclose area).
Checks:
- This graph: \(4 - 5 + 2 = 1\).
- Strang's second example: \(5 - 7 + 3 = 1\).
6. The framework of applied mathematics
Three steps link potentials to sources:
| Step | Equation | Name |
|---|---|---|
| Potentials → potential differences | \(e = Ax\) | |
| Potential differences → currents | \(y = Ce\) | Ohm's law. \(C\) is diagonal and holds the edge conductances, which are 1/resistance. |
| Currents → balance at nodes | \(A^\top y = f\) | KCL. \(f\) holds the external current sources, or \(f = 0\) if there are none. |
Combined into one equation:
Other places a source can enter. Batteries in the edges would enter at the \(e\) step instead.
Scope. Strang calls this the basic equation of applied mathematics in equilibrium: there is no time dependence.
Closing question. What do you know about \(A^\top A\), or \(A^\top C A\)? It is always symmetric:
The four subspaces, interpreted
| Subspace | Dimension | Meaning |
|---|---|---|
| \(N(A)\) | \(n - r = 1\) | Constant potentials, which drive no current |
| \(C(A^\top)\) | \(r = n - 1 = 3\) | Rows from edges forming a tree |
| \(N(A^\top)\) | \(m - r = 2\) | Currents around loops, satisfying KCL |
| \(C(A)\) | \(r = 3\) | Achievable potential differences |
A note on \(C(A)\). Strang does not develop \(C(A)\) in this lecture. A standard fact, not stated in the lecture: \(e \in C(A)\) exactly when the potential differences sum to zero around every loop. This is Kirchhoff's Voltage Law. It is also the orthogonality of \(C(A)\) and \(N(A^\top)\), which comes up in Lecture 14.
Big ideas from Lecture 12
- Incidence matrix. One row per edge, with entries \(-1\) and \(+1\). It is sparse and highly structured.
- \(N(A)\) is spanned by \((1,1,\dots,1)\). Potentials are determined only up to a constant, so the rank is \(n - 1\).
- \(N(A^\top)\) consists of the solutions of Kirchhoff's Current Law. Its basis is currents around independent loops.
- Independent edges form a tree, which has no loops.
- Euler's formula \(n - m + \ell = 1\) follows from \(\dim N(A^\top) = m - r\).
- Equilibrium equation \(A^\top C A x = f\), and the matrix \(A^\top C A\) is symmetric.