 5.2.1: For 15, determine whether the given set of vectors is an orthogonal...
 5.2.2: For 15, determine whether the given set of vectors is an orthogonal...
 5.2.3: For 15, determine whether the given set of vectors is an orthogonal...
 5.2.4: For 15, determine whether the given set of vectors is an orthogonal...
 5.2.5: For 15, determine whether the given set of vectors is an orthogonal...
 5.2.6: Let v = (7, 2). Determine all nonzero vectors w in R2 such that {v,...
 5.2.7: Let v = (3, 6, 1). Determine all vectors in R3 that are orthogonal ...
 5.2.8: Let v1 = (1, 2, 3), v2 = (1, 1, 1). Determine all nonzero vectors w...
 5.2.9: Let v1 = (4, 0, 0, 1), v2 = (1, 2, 3, 4). Determine all vectors in ...
 5.2.10: For 1012, show that the given set of vectors is an orthogonal set i...
 5.2.11: For 1012, show that the given set of vectors is an orthogonal set i...
 5.2.12: For 1012, show that the given set of vectors is an orthogonal set i...
 5.2.13: Consider the vectors v = (1i, 1+2i), w = (2+i,z) in C2. Determine t...
 5.2.14: For 1416, show that the given functions in C0[1, 1] are orthogonal,...
 5.2.15: For 1416, show that the given functions in C0[1, 1] are orthogonal,...
 5.2.16: For 1416, show that the given functions in C0[1, 1] are orthogonal,...
 5.2.17: For 1718, show that the given functions are orthonormal on [1, 1].
 5.2.18: For 1718, show that the given functions are orthonormal on [1, 1].
 5.2.19: Let p(x) = 2 x x2 and q(x) = 1 + x + x2. Using the inner product a0...
 5.2.20: Let A1 = 1 1 1 2 , A2 = 1 1 2 1 , and A3 = 1 3 0 2 . Use the inner ...
 5.2.21: Consider the problem of finding the distance from a point P(x0, y0)...
 5.2.22: For 2227, find the distance from the given point P to the given lin...
 5.2.23: For 2227, find the distance from the given point P to the given lin...
 5.2.24: For 2227, find the distance from the given point P to the given lin...
 5.2.25: For 2227, find the distance from the given point P to the given lin...
 5.2.26: For 2227, find the distance from the given point P to the given lin...
 5.2.27: For 2227, find the distance from the given point P to the given lin...
 5.2.28: In this problem, we use the ideas of this section to derive a formu...
 5.2.29: For 2932, use the result of to find the distance from the given poi...
 5.2.30: For 2932, use the result of to find the distance from the given poi...
 5.2.31: For 2932, use the result of to find the distance from the given poi...
 5.2.32: For 2932, use the result of to find the distance from the given poi...
 5.2.33: Let {u1, u2, v} be linearly independent vectors in an inner product...
 5.2.34: Prove that if {v1, v2,..., vk } is an orthogonal set of vectors in ...
 5.2.35: The subject of Fourier series is concerned with the representation ...
Solutions for Chapter 5.2: Orthogonal Sets of Vectors and Orthogonal Projections
Full solutions for Differential Equations  4th Edition
ISBN: 9780321964670
Solutions for Chapter 5.2: Orthogonal Sets of Vectors and Orthogonal Projections
Get Full SolutionsThis textbook survival guide was created for the textbook: Differential Equations, edition: 4. Since 35 problems in chapter 5.2: Orthogonal Sets of Vectors and Orthogonal Projections have been answered, more than 21320 students have viewed full stepbystep solutions from this chapter. Differential Equations was written by and is associated to the ISBN: 9780321964670. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 5.2: Orthogonal Sets of Vectors and Orthogonal Projections includes 35 full stepbystep solutions.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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.

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.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!