- Appendix.1: In Exercises 14, use mathematical induction to prove that the formu...
- Appendix.2: In Exercises 14, use mathematical induction to prove that the formu...
- Appendix.3: In Exercises 14, use mathematical induction to prove that the formu...
- Appendix.4: In Exercises 14, use mathematical induction to prove that the formu...
- Appendix.5: In Exercises 5 and 6, propose a formula for the sum of the first te...
- Appendix.6: In Exercises 5 and 6, propose a formula for the sum of the first te...
- Appendix.7: In Exercises 7 and 8, use mathematical induction to prove the inequ...
- Appendix.8: In Exercises 7 and 8, use mathematical induction to prove the inequ...
- Appendix.9: Prove that for all integers
- Appendix.10: (From Chapter 2) Use mathematical induction to prove that assuming ...
- Appendix.11: (From Chapter 3) Use mathematical induction to prove that where are...
- Appendix.12: If is an integer and is odd, then is odd. (Hint: An odd number can ...
- Appendix.13: If and are real numbers and then
- Appendix.14: If and are real numbers such that and then
- Appendix.15: If and are real numbers and then
- Appendix.16: If a and b are real numbers and _a _ b_2 _ a2 _ b2, then a _ 0 or b...
- Appendix.17: If is a real number and then
- Appendix.18: Use proof by contradiction to prove that the sum of a rational numb...
- Appendix.19: (From Chapter 4) Use proof by contradiction to prove that in a give...
- Appendix.20: (From Chapter 4) Let be a linearly independent set. Use proof by co...
- Appendix.21: If and are real numbers and then
- Appendix.22: The product of two irrational numbers is irrational.
- Appendix.23: If and are real numbers such that and then
- Appendix.24: If is a polynomial function and then
- Appendix.25: If and are differentiable functions and then
- Appendix.26: (From Chapter 2) If and are matrices and then
- Appendix.27: (From Chapter 3) If is a matrix, then det_A_1_
Solutions for Chapter Appendix: Mathematical Induction and Other Forms of Proofs
Full solutions for Elementary Linear Algebra | 6th Edition
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
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.
Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
Invert A by row operations on [A I] to reach [I A-I].
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.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
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.
Nullspace matrix N.
The columns of N are the n - r special solutions to As = O.
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
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·
Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).