Incidence Matrix each row describes from node i to node j the direction (node departed is -1, node arrived at is 1)
for a closed loop of a graph the rows of the matrix are linearly dependent
In the incidence matrix rows represent the flow of current of an edge (-1 means from that edge and +1 means to that edge) and the columns represent the currents flowing to and from nodes
x1 x2 x3 x4 are potentials at the nodes Since rows of A represent the directions of current of the edge, multiplying A to vector x represents the multiplication of potentials of the nodes (x1,x2,x3,x4) to the respective columns (that represent the how many currents are flowing into the node or out of the node ) with addition of all columns finally representing the potential difference of the edges
if all (x1,x2,x3,x4 …) are same the potential difference vector will be zero the Solution of null space would be as shown in above image dimension of null space 1 dim
dim of null space ⇒1=n-r⇒1=4-r⇒r=3
Dim of column space =3
Any of the three columns of A are basis of Column space (because it is three if it was 2 we couldn’t have said any 2 columns)
Since rows of At represent how many currents are flowing into the node or out of the node multiplying At with y vector of currents of the edges (potential difference vector divided by say the resistance to represent the current flowing through the edges) results in zeros as the current inflow must equal to current outflow
Each vector y which satisfies this equation represents a configuration of current flow through the graph network where current is flowing in a closed loop
Unique solutions must represent the no of unique closed loop configurations which can be used to represent every possible solution configuration of the graph network
dimension of null space of At = no of unique closed loops
directly looking at the graph the solution currents can be found following a path of trying to form a closed loop