- 5.5.1: In C[a% b], define the product</*> = [ f(t)g{t)dt. J a Show that th...
- 5.5.2: Does the equation</. g + A) = < /.g )+ </.*> hold for all elements ...
- 5.5.3: Consider a matrix S in Rnxn. In R", define the product(x.y) = {Sx)T...
- 5.5.4: In R" xm, consider the inner product(A, B) = trace(/\7 #) defined i...
- 5.5.5: Is ((/A, B)) = trace(^fi^) an inner product in Rwxm? (The notation ...
- 5.5.6: a. Consider an n x m matrix P and an m x n matrix Q. Show thattrace...
- 5.5.7: Consider an inner product (v,w) in a space V, and a scalar k. For w...
- 5.5.8: Consider an inner product (v. w) in a space V. Let w be a fixed ele...
- 5.5.9: Recall that a function f(t) from R to R is calledeven if f ( t) = /...
- 5.5.10: Consider the space P2 with inner producti r' (f.g) = j J ( f(t)g(t)...
- 5.5.11: The angle between two nonzero elements i> and w of an inner product...
- 5.5.12: Find all Fourier coefficients of the absolute value functionf(t) = ui
- 5.5.13: For a function / in C[n, n] (with the inner product defined on page...
- 5.5.14: Which of the following is an inner product in P2? Explain. a. (/,*)...
- 5.5.15: For which values of the constants /?, c, and d is the following an ...
- 5.5.16: a. Find an orthonormal basis of the space P\ with inner product(/ J...
- 5.5.17: Consider a linear space V. For which linear transforma- tions T fro...
- 5.5.18: Consider an orthonormal basis si^ of the inner product space V. For...
- 5.5.19: For which 2 x 2 matrices A is(5, w) = vTAwan inner product in R2 ? ...
- 5.5.20: Consider the inner product(v, w) = vTin R2. (See Exercise 19.)a. Fi...
- 5.5.21: If || 51| denotes the standard norm in IR", does the formula(v, w) ...
- 5.5.22: If f(t) is a continuous function, what is the relationship between ...
- 5.5.23: In the space P| of the polynomials of degree < 1, we define the inn...
- 5.5.24: Consider the linear space P of all polynomials, with inner productr...
- 5.5.25: Find the norm \\x || of(t2 is defined in Example 2.)
- 5.5.26: Find the Fourier coefficients of the piecewise continuous function/...
- 5.5.27: Find the Fourier coefficients of the piecewise continuous functionr...
- 5.5.28: Apply Theorem 5.5.6 to your answer in Exercise 26.
- 5.5.29: Apply Theorem 5.5.6 to your answer in Exercise 27.
- 5.5.30: Consider an ellipse E in R2 centered at the origin. Show that there...
- 5.5.31: Gaussian integration. In an introductory calculus course, you may h...
- 5.5.32: In the space C[ 1, 1], we introduce the inner product{/g) = \ / , f...
- 5.5.33: a. Let w(t) be a positive-valued function in C[a, b]t where b > a. ...
- 5.5.34: In the space Cf1,1], we define the inner product (/,> = /_!, - t2f(...
- 5.5.35: In this exercise, we compare the inner products and norms introduce...
Solutions for Chapter 5.5: Inner Product Spaces
Full solutions for Linear Algebra with Applications | 4th Edition
ISBN: 9780136009269
Solutions for Chapter 5.5: Inner Product Spaces
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Solutions for Chapter 5.5
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This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 4. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009269. Chapter 5.5: Inner Product Spaces includes 35 full step-by-step solutions. Since 35 problems in chapter 5.5: Inner Product Spaces have been answered, more than 63828 students have viewed full step-by-step solutions from this chapter.
Key Math Terms and definitions covered in this textbook
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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).
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Column space C (A) =
space of all combinations of the columns of A.
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Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
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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
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Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
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Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
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Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
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Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
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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.
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Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
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Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
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Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
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Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
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Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
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Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
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Solvable system Ax = b.
The right side b is in the column space of A.
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Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
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Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
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Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
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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.