- 4.5.1E: For each subspace in Exercises 1–8, (a) find a basis for the subspa...
- 4.5.2E: For each subspace in Exercises 1–8, (a) find a basis for the subspa...
- 4.5.3E: For each subspace in Exercises 1–8, (a) find a basis for the subspa...
- 4.5.4E: For each subspace in Exercises 1–8, (a) find a basis for the subspa...
- 4.5.5E: For each subspace in Exercises 1–8, (a) find a basis for the subspa...
- 4.5.6E: For each subspace in Exercises 1–8, (a) find a basis for the subspa...
- 4.5.7E: For each subspace in Exercises 1–8, (a) find a basis for the subspa...
- 4.5.8E: For each subspace in Exercises 1–8, (a) find a basis for the subspa...
- 4.5.9E: Find the dimension of the subspace of all vectors in whose first an...
- 4.5.10E: Find the dimension of the subspace H of spanned by
- 4.5.11E: In Exercises 11 and 12, find the dimension of the subspace spanned ...
- 4.5.12E: In Exercises 11 and 12, find the dimension of the subspace spanned ...
- 4.5.13E: Determine the dimensions of Nul A and Col A for the matrices shown ...
- 4.5.14E: Determine the dimensions of Nul A and Col A for the matrices shown ...
- 4.5.15E: Determine the dimensions of Nul A and Col A for the matrices shown ...
- 4.5.16E: Determine the dimensions of Nul A and Col A for the matrices shown ...
- 4.5.17E: Determine the dimensions of Nul A and Col A for the matrices shown ...
- 4.5.18E: Determine the dimensions of Nul A and Col A for the matrices shown ...
- 4.5.19E: In Exercises 19 and 20, V is a vector space. Mark each statement Tr...
- 4.5.20E: In Exercises 19 and 20, V is a vector space. Mark each statement Tr...
- 4.5.21E: The first four Hermite polynomials are 1, 2t, -2 + 4t2 and -12t + 8...
- 4.5.22E: The first four Laguerre polynomials are 1, 1 - t, 2 - 4t + t2, and ...
- 4.5.23E: Let B be the basis of consisting of the Hermite polynomials in Exer...
- 4.5.24E: Let be the basis of P2 consisting of the first three Laguerre polyn...
- 4.5.25E: Let S be a subset of an n-dimensional vector space V, and suppose S...
- 4.5.26E: Let H be an n-dimensional subspace of an n-dimensional vector space...
- 4.5.27E: Explain why the space of all polynomials is an infinite-dimensional...
- 4.5.28E: Show that the space of all continuous functions defined on the real...
- 4.5.29E: In Exercises 29 and 30, V is a nonzero finite-dimensional vector sp...
- 4.5.30E: In Exercises 29 and 30, V is a nonzero finite-dimensional vector sp...
- 4.5.31E: Exercises 31 and 32 concern finite-dimensional vector spaces V and ...
- 4.5.32E: Exercises 31 and 32 concern finite-dimensional vector spaces V and ...
- 4.5.33E: [M] According to Theorem 11, a linearly independent set in can be e...
- 4.5.34E: Assume the following trigonometric identities (see Exercise 37 in S...
Solutions for Chapter 4.5: Linear Algebra and Its Applications 4th Edition
Full solutions for Linear Algebra and Its Applications | 4th Edition
ISBN: 9780321385178
Solutions for Chapter 4.5
15
1
Since 34 problems in chapter 4.5 have been answered, more than 28124 students have viewed full step-by-step solutions from this chapter. Chapter 4.5 includes 34 full step-by-step solutions. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications, edition: 4. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321385178. This expansive textbook survival guide covers the following chapters and their solutions.
Key Math Terms and definitions covered in this textbook
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Cayley-Hamilton Theorem.
peA) = det(A - AI) has peA) = zero matrix.
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Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
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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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Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
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Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
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Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
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Free columns of A.
Columns without pivots; these are combinations of earlier columns.
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Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
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Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
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lA-II = 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.
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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.
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Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
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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.
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Outer product uv T
= column times row = rank one matrix.
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Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
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Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
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Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
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Solvable system Ax = b.
The right side b is in the column space of A.
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Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.