 5.5.1: Finding Lengths, Dot Product, and Distance In Exercises 18, find (a...
 5.1: Finding Lengths, Dot Product, and Distance In Exercises 18, find (a...
 5.5.2: Finding Lengths, Dot Product, and Distance In Exercises 18, find (a...
 5.2: Finding Lengths, Dot Product, and Distance In Exercises 18, find (a...
 5.5.3: Finding Lengths, Dot Product, and Distance In Exercises 18, find (a...
 5.3: Finding Lengths, Dot Product, and Distance In Exercises 18, find (a...
 5.5.4: Finding Lengths, Dot Product, and Distance In Exercises 18, find (a...
 5.4: Finding Lengths, Dot Product, and Distance In Exercises 18, find (a...
 5.5.5: Finding Lengths, Dot Product, and Distance In Exercises 18, find (a...
 5.5: Finding Lengths, Dot Product, and Distance In Exercises 18, find (a...
 5.5.6: Finding Lengths, Dot Product, and Distance In Exercises 18, find (a...
 5.6: Finding Lengths, Dot Product, and Distance In Exercises 18, find (a...
 5.5.7: Finding Lengths, Dot Product, and Distance In Exercises 18, find (a...
 5.7: Finding Lengths, Dot Product, and Distance In Exercises 18, find (a...
 5.5.8: Finding Lengths, Dot Product, and Distance In Exercises 18, find (a...
 5.8: Finding Lengths, Dot Product, and Distance In Exercises 18, find (a...
 5.5.9: Finding Length and a Unit Vector In Exercises 912, find !v! and fin...
 5.9: Finding Length and a Unit Vector In Exercises 912, find !v! and fin...
 5.5.10: Finding Length and a Unit Vector In Exercises 912, find !v! and fin...
 5.10: Finding Length and a Unit Vector In Exercises 912, find !v! and fin...
 5.5.11: Finding Length and a Unit Vector In Exercises 912, find !v! and fin...
 5.11: Finding Length and a Unit Vector In Exercises 912, find !v! and fin...
 5.5.12: Finding Length and a Unit Vector In Exercises 912, find !v! and fin...
 5.12: Finding Length and a Unit Vector In Exercises 912, find !v! and fin...
 5.5.13: Consider the vector v = (8, 8, 6). Find u such that(a) u has the sa...
 5.13: Consider the vector v = (8, 8, 6). Find u such that(a) u has the sa...
 5.5.14: For what values of c is !c(2, 2, 1)! = 3?
 5.14: For what values of c is !c(2, 2, 1)! = 3?
 5.5.15: Finding the Angle Between Two Vectors In Exercises 1520, find the a...
 5.15: Finding the Angle Between Two Vectors In Exercises 1520, find the a...
 5.5.16: Finding the Angle Between Two Vectors In Exercises 1520, find the a...
 5.16: Finding the Angle Between Two Vectors In Exercises 1520, find the a...
 5.5.17: Finding the Angle Between Two Vectors In Exercises 1520, find the a...
 5.17: Finding the Angle Between Two Vectors In Exercises 1520, find the a...
 5.5.18: Finding the Angle Between Two Vectors In Exercises 1520, find the a...
 5.18: Finding the Angle Between Two Vectors In Exercises 1520, find the a...
 5.5.19: Finding the Angle Between Two Vectors In Exercises 1520, find the a...
 5.19: Finding the Angle Between Two Vectors In Exercises 1520, find the a...
 5.5.20: Finding the Angle Between Two Vectors In Exercises 1520, find the a...
 5.20: Finding the Angle Between Two Vectors In Exercises 1520, find the a...
 5.5.21: Finding Orthogonal Vectors In Exercises 2124, determine all vectors...
 5.21: Finding Orthogonal Vectors In Exercises 2124, determine all vectors...
 5.5.22: Finding Orthogonal Vectors In Exercises 2124, determine all vectors...
 5.22: Finding Orthogonal Vectors In Exercises 2124, determine all vectors...
 5.5.23: Finding Orthogonal Vectors In Exercises 2124, determine all vectors...
 5.23: Finding Orthogonal Vectors In Exercises 2124, determine all vectors...
 5.5.24: Finding Orthogonal Vectors In Exercises 2124, determine all vectors...
 5.24: Finding Orthogonal Vectors In Exercises 2124, determine all vectors...
 5.5.25: For u = (4, 32, 1) and v = (12, 3, 1), (a) find the innerproduct re...
 5.25: For u = (4, 32, 1) and v = (12, 3, 1), (a) find the innerproduct re...
 5.5.26: For u = (0, 3, 13) and v = (43, 1, 3), (a) find the innerproduct re...
 5.26: For u = (0, 3, 13) and v = (43, 1, 3), (a) find the innerproduct re...
 5.5.27: Verify the triangle inequality and the CauchySchwarz Inequality fo...
 5.27: Verify the triangle inequality and the CauchySchwarz Inequality fo...
 5.5.28: Verify the triangle inequality and the CauchySchwarz Inequality fo...
 5.28: Verify the triangle inequality and the CauchySchwarz Inequality fo...
 5.5.29: Calculus In Exercises 29 and 30, (a) find the inner product, (b) de...
 5.29: Calculus In Exercises 29 and 30, (a) find the inner product, (b) de...
 5.5.30: Calculus In Exercises 29 and 30, (a) find the inner product, (b) de...
 5.30: Calculus In Exercises 29 and 30, (a) find the inner product, (b) de...
 5.5.31: Finding an Orthogonal Projection In Exercises 3136, find projvu.u =...
 5.31: Finding an Orthogonal Projection In Exercises 3136, find projvu.u =...
 5.5.32: Finding an Orthogonal Projection In Exercises 3136, find projvu.u =...
 5.32: Finding an Orthogonal Projection In Exercises 3136, find projvu.u =...
 5.5.33: Finding an Orthogonal Projection In Exercises 3136, find projvu.u =...
 5.33: Finding an Orthogonal Projection In Exercises 3136, find projvu.u =...
 5.5.34: Finding an Orthogonal Projection In Exercises 3136, find projvu.u =...
 5.34: Finding an Orthogonal Projection In Exercises 3136, find projvu.u =...
 5.5.35: Finding an Orthogonal Projection In Exercises 3136, find projvu.u =...
 5.35: Finding an Orthogonal Projection In Exercises 3136, find projvu.u =...
 5.5.36: Finding an Orthogonal Projection In Exercises 3136, find projvu.u =...
 5.36: Finding an Orthogonal Projection In Exercises 3136, find projvu.u =...
 5.5.37: Applying the GramSchmidt Process In Exercises 3740, apply the Gram...
 5.37: Applying the GramSchmidt Process In Exercises 3740, apply the Gram...
 5.5.38: Applying the GramSchmidt Process In Exercises 3740, apply the Gram...
 5.38: Applying the GramSchmidt Process In Exercises 3740, apply the Gram...
 5.5.39: Applying the GramSchmidt Process In Exercises 3740, apply the Gram...
 5.39: Applying the GramSchmidt Process In Exercises 3740, apply the Gram...
 5.5.40: Applying the GramSchmidt Process In Exercises 3740, apply the Gram...
 5.40: Applying the GramSchmidt Process In Exercises 3740, apply the Gram...
 5.5.41: Let B = {(0, 2, 2), (1, 0, 2)} be a basis for asubspace of R3, and ...
 5.41: Let B = {(0, 2, 2), (1, 0, 2)} be a basis for asubspace of R3, and ...
 5.5.42: Repeat Exercise 41 for B = {(1, 2, 2), (1, 0, 0)} and x = (3, 4, 4).
 5.42: Repeat Exercise 41 for B = {(1, 2, 2), (1, 0, 0)} and x = (3, 4, 4).
 5.5.43: Calculus In Exercises 4346, let f and g be functionsin the vector s...
 5.43: Calculus In Exercises 4346, let f and g be functionsin the vector s...
 5.5.44: Calculus In Exercises 4346, let f and g be functionsin the vector s...
 5.44: Calculus In Exercises 4346, let f and g be functionsin the vector s...
 5.5.45: Calculus In Exercises 4346, let f and g be functionsin the vector s...
 5.45: Calculus In Exercises 4346, let f and g be functionsin the vector s...
 5.5.46: Calculus In Exercises 4346, let f and g be functionsin the vector s...
 5.46: Calculus In Exercises 4346, let f and g be functionsin the vector s...
 5.5.47: Find an orthonormal basis for the subspace of Euclidean3space belo...
 5.47: Find an orthonormal basis for the subspace of Euclidean3space belo...
 5.5.48: Find an orthonormal basis for the solution space of thehomogeneous ...
 5.48: Find an orthonormal basis for the solution space of thehomogeneous ...
 5.5.49: Proof Prove that if u, v, and w are vectors in Rn, then(u + v) w = ...
 5.49: Proof Prove that if u, v, and w are vectors in Rn, then(u + v) w = ...
 5.5.50: Proof Prove that if u and v are vectors in Rn, then!u + v!2 + !u v!...
 5.50: Proof Prove that if u and v are vectors in Rn, then!u + v!2 + !u v!...
 5.5.51: Proof Prove that if u and v are vectors in an innerproduct space su...
 5.51: Proof Prove that if u and v are vectors in an innerproduct space su...
 5.5.52: Proof Prove that if u and v are vectors in an innerproduct space V,...
 5.52: Proof Prove that if u and v are vectors in an innerproduct space V,...
 5.5.53: Proof Let V be an mdimensional subspace of Rnsuch that m < n. Prov...
 5.53: Proof Let V be an mdimensional subspace of Rnsuch that m < n. Prov...
 5.5.54: Let V be the twodimensional subspace of R4 spannedby (0, 1, 0, 1) ...
 5.54: Let V be the twodimensional subspace of R4 spannedby (0, 1, 0, 1) ...
 5.5.55: Proof Let {u1, u2, . . . , um} be an orthonormal subsetof Rn, and l...
 5.55: Proof Let {u1, u2, . . . , um} be an orthonormal subsetof Rn, and l...
 5.5.56: Proof Let {x1, x2, . . . , xn} be a set of real numbers.Use the Cau...
 5.56: Proof Let {x1, x2, . . . , xn} be a set of real numbers.Use the Cau...
 5.5.57: Proof Let u and v be vectors in an inner productspace V. Prove that...
 5.57: Proof Let u and v be vectors in an inner productspace V. Prove that...
 5.5.58: Writing Let {u1, u2, . . . , un} be a dependent set ofvectors in an...
 5.58: Writing Let {u1, u2, . . . , un} be a dependent set ofvectors in an...
 5.5.59: Find the orthogonal complement S of the subspace Sof R3 spanned by ...
 5.59: Find the orthogonal complement S of the subspace Sof R3 spanned by ...
 5.5.60: Find the projection of the vector v = [1 0 2]Tonto the subspaceS = ...
 5.60: Find the projection of the vector v = [1 0 2]Tonto the subspaceS = ...
 5.5.61: Find bases for the four fundamental subspaces of thematrixA = [0011...
 5.61: Find bases for the four fundamental subspaces of thematrixA = [0011...
 5.5.62: Find the least squares regression line for the set of datapoints{(2...
 5.62: Find the least squares regression line for the set of datapoints{(2...
 5.5.63: Revenue The table shows the revenues y (in billionsof dollars) for ...
 5.63: Revenue The table shows the revenues y (in billionsof dollars) for ...
 5.5.64: Petroleum Production The table shows the North American petroleum p...
 5.64: Petroleum Production The table shows the North American petroleum p...
 5.5.65: Finding the Cross Product In Exercises 6568, find u v and show that...
 5.65: Finding the Cross Product In Exercises 6568, find u v and show that...
 5.5.66: Finding the Cross Product In Exercises 6568, find u v and show that...
 5.66: Finding the Cross Product In Exercises 6568, find u v and show that...
 5.5.67: Finding the Cross Product In Exercises 6568, find u v and show that...
 5.67: Finding the Cross Product In Exercises 6568, find u v and show that...
 5.5.68: Finding the Cross Product In Exercises 6568, find u v and show that...
 5.68: Finding the Cross Product In Exercises 6568, find u v and show that...
 5.5.69: Finding the Volume of a Parallelepiped In Exercises6972, find the v...
 5.69: Finding the Volume of a Parallelepiped In Exercises6972, find the v...
 5.5.70: Finding the Volume of a Parallelepiped In Exercises6972, find the v...
 5.70: Finding the Volume of a Parallelepiped In Exercises6972, find the v...
 5.5.71: Finding the Volume of a Parallelepiped In Exercises6972, find the v...
 5.71: Finding the Volume of a Parallelepiped In Exercises6972, find the v...
 5.5.72: Finding the Volume of a Parallelepiped In Exercises6972, find the v...
 5.72: Finding the Volume of a Parallelepiped In Exercises6972, find the v...
 5.5.73: Find the area of the parallelogram that hasu = (1, 3, 0) and v = (1...
 5.73: Find the area of the parallelogram that hasu = (1, 3, 0) and v = (1...
 5.5.74: Proof Prove that!u v! = !u! !v!if and only if u and v are orthogonal.
 5.74: Proof Prove that!u v! = !u! !v!if and only if u and v are orthogonal.
 5.5.75: Finding a Least Squares Approximation In Exercises 7578, (a) find t...
 5.75: Finding a Least Squares Approximation In Exercises 7578, (a) find t...
 5.5.76: Finding a Least Squares Approximation In Exercises 7578, (a) find t...
 5.76: Finding a Least Squares Approximation In Exercises 7578, (a) find t...
 5.5.77: Finding a Least Squares Approximation In Exercises 7578, (a) find t...
 5.77: Finding a Least Squares Approximation In Exercises 7578, (a) find t...
 5.5.78: Finding a Least Squares Approximation In Exercises 7578, (a) find t...
 5.78: Finding a Least Squares Approximation In Exercises 7578, (a) find t...
 5.5.79: Finding a Least Squares Approximation In Exercises 79 and 80, (a) f...
 5.79: Finding a Least Squares Approximation In Exercises 79 and 80, (a) f...
 5.5.80: Finding a Least Squares Approximation In Exercises 79 and 80, (a) f...
 5.80: Finding a Least Squares Approximation In Exercises 79 and 80, (a) f...
 5.5.81: Finding a Fourier Approximation In Exercises 81 and 82, find the Fo...
 5.81: Finding a Fourier Approximation In Exercises 81 and 82, find the Fo...
 5.5.82: Finding a Fourier Approximation In Exercises 81 and 82, find the Fo...
 5.82: Finding a Fourier Approximation In Exercises 81 and 82, find the Fo...
 5.5.83: True or False? In Exercises 83 and 84, determine whether each state...
 5.83: True or False? In Exercises 83 and 84, determine whether each state...
 5.5.84: True or False? In Exercises 83 and 84, determine whether each state...
 5.84: True or False? In Exercises 83 and 84, determine whether each state...
Solutions for Chapter 5: Inner Product Spaces
Full solutions for Elementary Linear Algebra  8th Edition
ISBN: 9781305658004
Solutions for Chapter 5: Inner Product Spaces
Get Full SolutionsSince 168 problems in chapter 5: Inner Product Spaces have been answered, more than 44814 students have viewed full stepbystep solutions from this chapter. Chapter 5: Inner Product Spaces includes 168 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 8. This expansive textbook survival guide covers the following chapters and their solutions. Elementary Linear Algebra was written by and is associated to the ISBN: 9781305658004.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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.

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

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.

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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.