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# Solutions for Chapter 6.4: Vector Space of Matrices and Vector Space of Linear Transformations (Optional)

## Full solutions for Elementary Linear Algebra with Applications | 9th Edition

ISBN: 9780132296540

Solutions for Chapter 6.4: Vector Space of Matrices and Vector Space of Linear Transformations (Optional)

Solutions for Chapter 6.4
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##### ISBN: 9780132296540

This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. Since 18 problems in chapter 6.4: Vector Space of Matrices and Vector Space of Linear Transformations (Optional) have been answered, more than 12141 students have viewed full step-by-step solutions from this chapter. Chapter 6.4: Vector Space of Matrices and Vector Space of Linear Transformations (Optional) includes 18 full step-by-step solutions. Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780132296540.

Key Math Terms and definitions covered in this textbook
• Back substitution.

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

• Circulant matrix C.

Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.

• Companion matrix.

Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).

• 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.

• 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.

• Iterative method.

A sequence of steps intended to approach the desired solution.

• 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.

• Krylov subspace Kj(A, b).

The subspace spanned by b, Ab, ... , Aj-Ib. 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.

• Linear transformation T.

Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

• Markov matrix M.

All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

• Network.

A directed graph that has constants Cl, ... , Cm associated with the edges.

• Pivot.

The diagonal entry (first nonzero) at the time when a row is used in elimination.

• 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.

• 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.

• Rank one matrix A = uvT f=. O.

Column and row spaces = lines cu and cv.

• Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.

Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

• Row space C (AT) = all combinations of rows of A.

Column vectors by convention.

• Spectrum of A = the set of eigenvalues {A I, ... , An}.

Spectral radius = max of IAi I.

• Volume of box.

The rows (or the columns) of A generate a box with volume I det(A) I.

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