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Get Full Access to Algebra - Textbook Survival Guide
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# Solutions for Chapter 2.2: MATRICES AND LINEAR TRANSFORMATIONS

## Full solutions for Elementary Linear Algebra: A Matrix Approach | 2nd Edition

ISBN: 9780131871410

Solutions for Chapter 2.2: MATRICES AND LINEAR TRANSFORMATIONS

Solutions for Chapter 2.2
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##### ISBN: 9780131871410

Elementary Linear Algebra: A Matrix Approach was written by and is associated to the ISBN: 9780131871410. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.2: MATRICES AND LINEAR TRANSFORMATIONS includes 26 full step-by-step solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra: A Matrix Approach, edition: 2. Since 26 problems in chapter 2.2: MATRICES AND LINEAR TRANSFORMATIONS have been answered, more than 21519 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
• 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.)

• Cross product u xv in R3:

Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

• 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.

• Echelon matrix U.

The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

• Elimination matrix = Elementary matrix Eij.

The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.

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

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

• Full column rank r = n.

Independent columns, N(A) = {O}, no free variables.

• 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.

• Left inverse A+.

If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

• Linearly dependent VI, ... , Vn.

A combination other than all Ci = 0 gives L Ci Vi = O.

• Multiplication Ax

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

• Pivot.

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

• Projection p = a(aTblaTa) onto the line through a.

P = aaT laTa has rank l.

• 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).

• 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 ().

• Saddle point of I(x}, ... ,xn ).

A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

• 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.

• Transpose matrix AT.

Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.

• Vandermonde matrix V.

V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.

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