- 2.2.1: If A b a Lc~lic matrix and ,. is a population di,tribution. then ea...
- 2.2.2: For any population disuibution v, if tl is the Leslie matrix for th...
- 2.2.3: In a Lc~lie matrix. the (i .j)entry equals the average num ber of f...
- 2.2.4: n a l..c>lie matrix. the (i + I. i)-entry cquab the portion of fema...
- 2.2.5: The applicalion in I his sec1ion on traffic flow relies on the asso...
- 2.2.6: If A and 8 are matrices and x, y. and z are vectors such I. If A b ...
- 2.2.7: A (0. 1)-malrix is a matrix with Os and Is as its only e nlries
- 2.2.8: A (0, I )-matrix is a square matrix with Os and Is as its only entries
- 2.2.9: Every (0, I )matrix is a symmetric malrix.
- 2.2.10: By observing a certain colony of mice. reearchcrs found that all an...
- 2.2.11: Suppose that the females of a certain colony of animals arc divided...
- 2.2.12: A certain colony of lizards has a life span of less than 3 years. S...
- 2.2.13: A certain colony of bat> has a life span of less than 3 years. Supp...
- 2.2.14: A certain colony of voles has a life span of less than 3 years. Sup...
- 2.2.15: A cenain colony of squirrels has a life span of less than 3 years. ...
- 2.2.16: The maximum membership term for each member of the Service Club is ...
- 2.2.17: A cenain medical foundation rccetves money from two sources: donatt...
- 2.2.18: Water is pumped into a system of pipes at points P1 and Pz shown in...
- 2.2.19: With the interpretation of a (0. I )-matrix found in Prnctice 3. ~u...
- 2.2.20: Recall the (0. 1 )-matrix ~ [l 0 l1 0 0 l 0 0 0 0 0 I 0 in which th...
- 2.2.21: Suppose that there is a group of four people and an associated -' x...
- 2.2.22: Suppo-c that student preference for a ;.et of courses is given in t...
- 2.2.23: Let A be the matrix in E'ample 2. (a) Justify the following interpr...
- 2.2.24: Sup1X>se that A and 8 arc m x m matrices that commute: that is. AB ...
- 2.2.25: In reference to the apphcallon in the text involving the N:1tchel I...
- 2.2.26: Suppose that we have a group of six people. each of whom own< a com...
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
29
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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
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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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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].
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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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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.
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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.
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Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
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Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.
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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.
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Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
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Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
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Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.
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Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
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Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
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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).
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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 ().
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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.
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Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
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Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
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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.
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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.