 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
ISBN: 9780321962218
Solutions for Chapter 5.4: Inner Product Spaces
Get Full SolutionsLinear Algebra with Applications was written by and is associated to the ISBN: 9780321962218. This expansive textbook survival guide covers the following chapters and their solutions. Since 33 problems in chapter 5.4: Inner Product Spaces have been answered, more than 10893 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 9. Chapter 5.4: Inner Product Spaces includes 33 full stepbystep solutions.

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!

Block matrix.
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.

Cholesky factorization
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 IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Fundamental Theorem.
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 AI.
Square matrix with AI A = I and AAl = 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 B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Iterative method.
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, ... , 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.

lAII = 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.

Lucas numbers
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.

Partial pivoting.
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.

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