- 2.6.1E: Let A = a) What size is A?________________b) What is the third colu...
- 2.6.2E: Find A + B. where ________________
- 2.6.3E: Find AB if ________________ ________________
- 2.6.4E: Find the product AB. where ________________ ________________
- 2.6.5E: Find a matrix A such that [Hint: Finding A requires that you solve ...
- 2.6.6E: Find a matrix A such that
- 2.6.7E: Let A be an m × n matrix and let 0 be the m × n matrix that has all...
- 2.6.8E: Show that matrix addition is commutative; that is, show that if A a...
- 2.6.9E: Show that matrix addition is associative: that is, show that if A, ...
- 2.6.10E: Let A be a 3 × 4 matrix, B be a 4 × 5 matrix, and C be a 4×4 matrix...
- 2.6.11E: What do we know about the sizes of the matrices A and B if both of ...
- 2.6.12E: In this exercise we show that matrix multiplication is distributive...
- 2.6.13E: In this exercise we show that matrix multiplication is associative....
- 2.6.14E: The n × n matrix A =[aij] is called a diagonal matrix i aij = 0 whe...
- 2.6.15E: Let Find a formula for An , whenever n is a positive integer.
- 2.6.16E: Show that (At)t = A.
- 2.6.17E: Let A and B be two n × n matrices. Show thata) (A + B)t = At +Bt.__...
- 2.6.20E: Let a) Find A?1. [Hint: Use Exercise 19.]________________b) Find A3...
- 2.6.21E: Let A be an invertible matrix. Show that (An)?l = (A?1) n whenever ...
- 2.6.22E: Let A be a matrix. Show that the matrix AAt is symmetric. |Hint: Sh...
- 2.6.23E: Suppose that A is an n × n matrix where n is a positive integer. Sh...
- 2.6.24E: a) Show that the system of simultaneous linear equation in the vari...
- 2.6.25E: Use Exercises 18 and 24 to solve the system
- 2.6.26E: Let Find ________________ ________________
- 2.6.27E: Let Find ________________ ________________
- 2.6.28E: Find the Boolean product of A and B. where
- 2.6.29E: Let Find ________________ ________________
- 2.6.30E: Let A be a zero–one matrix. Show thata) A ? A = A. ________________...
- 2.6.31E: In this exercise we show that the meet and join operations are comm...
- 2.6.32E: In this exercise we show that the meet and join operations are asso...
- 2.6.33E: We will establish distributive laws of the meet over the join opera...
- 2.6.34E: Let A be an n × n zero–one matrix. Let I be the n × n identity matr...
- 2.6.35E: In this exercise we will show that the Boolean product of zero–one ...
Solutions for Chapter 2.6: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications | 7th Edition
Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).
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.
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Gram-Schmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.
Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.
A sequence of steps intended to approach the desired solution.
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.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
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).
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.
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.