 5.1: In Exercises 18, find (a) (b) (c) and
 5.2: In Exercises 18, find (a) (b) (c) and
 5.3: In Exercises 18, find (a) (b) (c) and
 5.4: In Exercises 18, find (a) (b) (c) and
 5.5: In Exercises 18, find (a) (b) (c) and
 5.6: In Exercises 18, find (a) (b) (c) and
 5.7: In Exercises 18, find (a) (b) (c) and
 5.8: In Exercises 18, find (a) (b) (c) and
 5.9: In Exercises 912, find and find a unit vector in the direction of
 5.10: In Exercises 912, find and find a unit vector in the direction of
 5.11: In Exercises 912, find and find a unit vector in the direction of
 5.12: In Exercises 912, find and find a unit vector in the direction of
 5.13: In Exercises 1318, find the angle between u and v.
 5.14: In Exercises 1318, find the angle between u and v.
 5.15: In Exercises 1318, find the angle between u and v.
 5.16: In Exercises 1318, find the angle between u and v.
 5.17: In Exercises 1318, find the angle between u and v.
 5.18: In Exercises 1318, find the angle between u and v.
 5.19: In Exercises 1924, find
 5.20: In Exercises 1924, find
 5.21: In Exercises 1924, find
 5.22: In Exercises 1924, find
 5.23: In Exercises 1924, find
 5.24: In Exercises 1924, find
 5.25: For and (a) find the inner product represented by and (b) use this ...
 5.26: For and (a) find the inner product represented by and (b) use this ...
 5.27: Verify the Triangle Inequality and the CauchySchwarz Inequality fo...
 5.28: Verify the Triangle Inequality and the CauchySchwarz Inequality fo...
 5.29: In Exercises 2932, find all vectors orthogonal to u
 5.30: In Exercises 2932, find all vectors orthogonal to u
 5.31: In Exercises 2932, find all vectors orthogonal to u
 5.32: In Exercises 2932, find all vectors orthogonal to u
 5.33: In Exercises 3336, use the GramSchmidt orthonormalization process ...
 5.34: In Exercises 3336, use the GramSchmidt orthonormalization process ...
 5.35: In Exercises 3336, use the GramSchmidt orthonormalization process ...
 5.36: In Exercises 3336, use the GramSchmidt orthonormalization process ...
 5.37: Let be a basis for a subspace of and let be a vector in the subspac...
 5.38: Repeat Exercise 37 for and
 5.39: Calculus In Exercises 3942, let and be functions in the vector spac...
 5.40: Calculus In Exercises 3942, let and be functions in the vector spac...
 5.41: Calculus In Exercises 3942, let and be functions in the vector spac...
 5.42: Calculus In Exercises 3942, let and be functions in the vector spac...
 5.43: Find an orthonormal basis for the following subspace of Euclidean 3...
 5.44: Find an orthonormal basis for the solution space of the homogeneous...
 5.45: Calculus In Exercises 45 and 46, (a) find the inner product, (b) de...
 5.46: Calculus In Exercises 45 and 46, (a) find the inner product, (b) de...
 5.47: Prove that if and are vectors in an inner product space such that a...
 5.48: Prove that if and are vectors in an inner product space then
 5.49: Let be an mdimensional subspace of such that Prove that any vector...
 5.50: Let V be the twodimensional subspace of spanned by and Write the v...
 5.51: Let be an orthonormal subset of and let be any vector in Prove that...
 5.52: Let be a set of real numbers. Use the CauchySchwarz Inequality to ...
 5.53: Let and be vectors in an inner product space Prove that if and only...
 5.54: Writing Let be a dependent set of vectors in an inner product space...
 5.55: Find the orthogonal complement of the subspace of spanned by the tw...
 5.56: Find the projection of the vector onto the subspace
 5.57: Find bases for the four fundamental subspaces of the matrix
 5.58: Find the least squares regression line for the set of data points G...
 5.59: The table shows the retail sales (in millions of dollars) of runnin...
 5.60: The table shows the average salaries (in thousands of dollars) for ...
 5.61: The table shows the world energy consumption (in quadrillions of Bt...
 5.62: The table shows the numbers of stores for the Target Corporation du...
 5.63: The table shows the revenues (in millions of dollars) for eBay, Inc...
 5.64: The table shows the revenues (in millions of dollars) for Google, I...
 5.65: The table shows the sales (in millions of dollars) for Circuit City...
 5.66: The Cross Product of Two Vectors in Space In Exercises 6669, find a...
 5.67: The Cross Product of Two Vectors in Space In Exercises 6669, find a...
 5.68: The Cross Product of Two Vectors in Space In Exercises 6669, find a...
 5.69: The Cross Product of Two Vectors in Space In Exercises 6669, find a...
 5.70: Find the area of the parallelogram that has and as adjacent sides.
 5.71: Prove that if and only if and are orthogonal.
 5.72: In Exercises 72 and 73, the volume of the parallelepiped having and...
 5.73: In Exercises 72 and 73, the volume of the parallelepiped having and...
 5.74: In Exercises 7477, find the linear least squares approximating func...
 5.75: In Exercises 7477, find the linear least squares approximating func...
 5.76: In Exercises 7477, find the linear least squares approximating func...
 5.77: In Exercises 7477, find the linear least squares approximating func...
 5.78: In Exercises 78 and 79, find the quadratic least squares approximat...
 5.79: In Exercises 78 and 79, find the quadratic least squares approximat...
 5.80: In Exercises 80 and 81, find the nthorder Fourier approximation of...
 5.81: In Exercises 80 and 81, find the nthorder Fourier approximation of...
 5.82: (a) The cross product of two nonzero vectors in yields a vector ort...
 5.83: (a) The vectors in and have equal lengths but opposite directions. ...
Solutions for Chapter 5: Inner Product Spaces
Full solutions for Elementary Linear Algebra  6th Edition
ISBN: 9780618783762
Solutions for Chapter 5: Inner Product Spaces
Get Full SolutionsSince 83 problems in chapter 5: Inner Product Spaces have been answered, more than 18105 students have viewed full stepbystep solutions from this chapter. Chapter 5: Inner Product Spaces includes 83 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 6. Elementary Linear Algebra was written by and is associated to the ISBN: 9780618783762. This expansive textbook survival guide covers the following chapters and their solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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

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).

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.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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!).

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

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.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Solvable system Ax = b.
The right side b is in the column space of A.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.