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# Solutions for Chapter 6.3: Matrices for Linear Transformations

## Full solutions for Elementary Linear Algebra | 6th Edition

ISBN: 9780618783762

Solutions for Chapter 6.3: Matrices for Linear Transformations

Solutions for Chapter 6.3
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##### ISBN: 9780618783762

Since 80 problems in chapter 6.3: Matrices for Linear Transformations have been answered, more than 43253 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 6.3: Matrices for Linear Transformations includes 80 full step-by-step 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.

Key Math Terms and definitions covered in this textbook
• Characteristic equation det(A - AI) = O.

The n roots are the eigenvalues of A.

• 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).

• Diagonalization

A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.

• Exponential eAt = I + At + (At)2 12! + ...

has derivative AeAt; eAt u(O) solves u' = Au.

• Full row rank r = m.

Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

• Identity matrix I (or In).

Diagonal entries = 1, off-diagonal entries = 0.

• 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.

• 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.

• Multiplication Ax

= Xl (column 1) + ... + xn(column n) = combination of columns.

• 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.

• Normal matrix.

If N NT = NT N, then N has orthonormal (complex) eigenvectors.

• Partial pivoting.

In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

• Permutation matrix P.

There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.

• 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.

• Rank r (A)

= number of pivots = dimension of column space = dimension of row space.

• Rotation matrix

R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().

• Schur complement S, D - C A -} B.

Appears in block elimination on [~ g ].

• Similar matrices A and B.

Every B = M-I AM has the same eigenvalues as A.

• Special solutions to As = O.

One free variable is Si = 1, other free variables = o.