- 5.4.1: Let x = (1, 1, 1, 1)T and y = (1, 1, 5, 3)T . Show that x y. Calcul...
- 5.4.2: Let x = (1, 1, 1, 1)T and y = (8, 2, 2, 0)T . (a) Determine the ang...
- 5.4.3: Use equation (1) with weight vector w = 1 4 , 1 2 , 1 4 T to define...
- 5.4.4: Given A = 122 102 311 and B = 411 332 1 2 2 determine the value of ...
- 5.4.5: Show that equation (2) defines an inner product on Rmn.
- 5.4.6: Show that the inner product defined by equation (3) satisfies the l...
- 5.4.7: In C[0, 1], with inner product defined by (3), compute (a) ex , ex ...
- 5.4.8: In C[0, 1], with inner product defined by (3), consider the vectors...
- 5.4.9: In C[, ] with inner product defined by (6), show that cos mx and si...
- 5.4.10: Show that the functions x and x2 are orthogonal in P5 with inner pr...
- 5.4.11: In P5 with inner product as in Exercise 10 and norm defined by p = ...
- 5.4.12: . If V is an inner product space, show that v = v, v satisfies the ...
- 5.4.13: . Show that x1 = n i=1 |xi| defines a norm on Rn. 1
- 5.4.14: Show that x = max 1in |xi| defines a norm on Rn. 1
- 5.4.15: Compute x1, x2, and x for each of the following vectors in R3. (a) ...
- 5.4.16: Let x = (5, 2, 4)T and y = (3, 3, 2)T . Compute x y1, x y2, and x y...
- 5.4.17: Let x and y be vectors in an inner product space. Show that if x y ...
- 5.4.18: Show that if u and v are vectors in an inner product space that sat...
- 5.4.19: Show that if u and v are vectors in an inner product space that sat...
- 5.4.20: Let A be a nonsingular n n matrix and for each vector x in Rn defin...
- 5.4.21: Let x Rn. Show that x x2. 22.
- 5.4.22: Let x R2. Show that x2 x1. [Hint: Write x in the form x1e1 + x2e2 a...
- 5.4.23: Give an example of a nonzero vector x R2 for which x = x2 = x1 24. Sh
- 5.4.24: Show that in any vector space with a norm v=v 25.
- 5.4.25: Show that for any u and v in a normed vector space u + v | uv | 26. Pr
- 5.4.26: Prove that, for any u and v in an inner product space V, u + v2 + u...
- 5.4.27: The result of Exercise 26 is not valid for norms other than the nor...
- 5.4.28: Determine whether the following define norms on C[a, b]: (a) f =| f...
- 5.4.29: Let x Rn and show that (a) x1 nx (b) x2 n x Give examples of vector...
- 5.4.30: . Sketch the set of points (x1, x2) = xT in R2 such that (a) x2 = 1...
- 5.4.31: Let K be an n n matrix of the form K = 1 c c c c 0 s sc sc sc 0 0 s...
- 5.4.32: The trace of an nn matrix C, denoted tr(C), is the sum of its diago...
- 5.4.33: Consider the vector space Rn with inner product x, y = xT y. Show t...
Solutions for Chapter 5.4: Inner Product Spaces
Full solutions for Linear Algebra with Applications | 9th Edition
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
A = CTC = (L.J]))(L.J]))T for positive definite A.
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
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.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
A sequence of steps intended to approach the desired solution.
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
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.
lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Solvable system Ax = b.
The right side b is in the column space of A.
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).