 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 ndimensional vector space V, and suppose S...
 4.5.26E: Let H be an ndimensional subspace of an ndimensional vector space...
 4.5.27E: Explain why the space of all polynomials is an infinitedimensional...
 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 finitedimensional vector sp...
 4.5.30E: In Exercises 29 and 30, V is a nonzero finitedimensional vector sp...
 4.5.31E: Exercises 31 and 32 concern finitedimensional vector spaces V and ...
 4.5.32E: Exercises 31 and 32 concern finitedimensional 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
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Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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.)

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.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

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.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

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 ().

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

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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.