- 3.5.1: Give the dimensions of the four fundamental subspaces of the incide...
- 3.5.2: Let A denote the incidence matrix of the disconnected graph shown i...
- 3.5.3: Give the dimensions of the four fundamental subspaces of the incide...
- 3.5.4: a. Show that in a graph with n nodes and n edges, there must be a l...
- 3.5.5: Ohms Law says that V = IR; that is, voltage (in volts) = current (i...
- 3.5.6: Use the approach of Exercise 5 to obtain the answer to Example 5 in...
Solutions for Chapter 3.5: A Graphic Example
Full solutions for Linear Algebra: A Geometric Approach | 2nd Edition
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].
Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
A directed graph that has constants Cl, ... , Cm associated with the edges.
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Outer product uv T
= column times row = rank one matrix.
Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.