 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
ISBN: 9780618783762
Solutions for Chapter Appendix: Mathematical Induction and Other Forms of Proofs
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Elementary Linear Algebra was written by and is associated to the ISBN: 9780618783762. Chapter Appendix: Mathematical Induction and Other Forms of Proofs includes 27 full stepbystep solutions. Since 27 problems in chapter Appendix: Mathematical Induction and Other Forms of Proofs have been answered, more than 12848 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 6.

Cyclic shift
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.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

GramSchmidt 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, ... , AjIb. 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.

Schwarz inequality
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 = UI.
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.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).