 2.5.1: In Exercises 1 12, compwe the product of each partitioned marrix u...
 2.5.2: In Exercises 1 12, compwe the product of each partitioned marrix u...
 2.5.3: In Exercises 1 12, compwe the product of each partitioned marrix u...
 2.5.4: In Exercises 1 12, compwe the product of each partitioned marrix u...
 2.5.5: In Exercises 1 12, compwe the product of each partitioned marrix u...
 2.5.6: In Exercises 1 12, compwe the product of each partitioned marrix u...
 2.5.7: In Exercises 1 12, compwe the product of each partitioned marrix u...
 2.5.8: In Exercises 1 12, compwe the product of each partitioned marrix u...
 2.5.9: In Exercises 1 12, compwe the product of each partitioned marrix u...
 2.5.10: In Exercises 1 12, compwe the product of each partitioned marrix u...
 2.5.11: In Exercises 1 12, compwe the product of each partitioned marrix u...
 2.5.12: In Exercises 1 12, compwe the product of each partitioned marrix u...
 2.5.13: In Exert:i.fe.s 13 20, compute the indicated row of the given prod...
 2.5.14: In Exert:i.fe.s 13 20, compute the indicated row of the given prod...
 2.5.15: In Exert:i.fe.s 13 20, compute the indicated row of the given prod...
 2.5.16: In Exert:i.fe.s 13 20, compute the indicated row of the given prod...
 2.5.17: In Exert:i.fe.s 13 20, compute the indicated row of the given prod...
 2.5.18: In Exert:i.fe.s 13 20, compute the indicated row of the given prod...
 2.5.19: In Exert:i.fe.s 13 20, compute the indicated row of the given prod...
 2.5.20: In Exert:i.fe.s 13 20, compute the indicated row of the given prod...
 2.5.21: In Exercises 21 28, use the mmrices A. B, and C from Exercises 13...
 2.5.22: In Exercises 21 28, use the mmrices A. B, and C from Exercises 13...
 2.5.23: In Exercises 21 28, use the mmrices A. B, and C from Exercises 13...
 2.5.24: In Exercises 21 28, use the mmrices A. B, and C from Exercises 13...
 2.5.25: In Exercises 21 28, use the mmrices A. B, and C from Exercises 13...
 2.5.26: In Exercises 21 28, use the mmrices A. B, and C from Exercises 13...
 2.5.27: In Exercises 21 28, use the mmrices A. B, and C from Exercises 13...
 2.5.28: In Exercises 21 28, use the mmrices A. B, and C from Exercises 13...
 2.5.29: The definition of the matrix product AB on page 97 can be regarded ...
 2.5.30: Let A and B be matrices such that AB is defined, and let A and B be...
 2.5.31: The outer product vw7 is defined only if v and w are both in'R.
 2.5.32: For any vectors v and win 'R."' and 'R.", re.~pectively, the outer ...
 2.5.33: For any vectors v and w in 'R."' and 'R.", respectively, the outer ...
 2.5.34: The product of an m x 11 nonzero matrix and an 11 x p nonzero matri...
 2.5.35: In Exercises 3540, assume thm A. 8. C, and D are 11 x nmmrices, 0 ...
 2.5.36: In Exercises 3540, assume thm A. 8. C, and D are 11 x nmmrices, 0 ...
 2.5.37: In Exercises 3540, assume thm A. 8. C, and D are 11 x nmmrices, 0 ...
 2.5.38: In Exercises 3540, assume thm A. 8. C, and D are 11 x nmmrices, 0 ...
 2.5.39: In Exercises 3540, assume thm A. 8. C, and D are 11 x nmmrices, 0 ...
 2.5.40: In Exercises 3540, assume thm A. 8. C, and D are 11 x nmmrices, 0 ...
 2.5.41: Show that if A. B, C. and D are 11 x 11 matrices such that A is inv...
 2.5.42: In Exercises 42 47, assume that A. 8 , C, and Dare 11 x 11 matrice...
 2.5.43: In Exercises 42 47, assume that A. 8 , C, and Dare 11 x 11 matrice...
 2.5.44: In Exercises 42 47, assume that A. 8 , C, and Dare 11 x 11 matrice...
 2.5.45: In Exercises 42 47, assume that A. 8 , C, and Dare 11 x 11 matrice...
 2.5.46: In Exercises 42 47, assume that A. 8 , C, and Dare 11 x 11 matrice...
 2.5.47: In Exercises 42 47, assume that A. 8 , C, and Dare 11 x 11 matrice...
 2.5.48: Let A and 8 be 11 x 11 matrices and 0 be the 11 x 11 zero matrix. U...
 2.5.49: Let A and 8 be 11 x 11 matrices and 0 be the 11 x 11 zero matrix. U...
 2.5.50: Let A and 8 be invertible 11 x 11 matrices. Prove that [ ~ ~] is in...
 2.5.51: Prove that if A and 8 arc invertible 11 x 11 matrices. then ~ ~] s ...
 2.5.52: Suppose a and b are nonzero vectors in n'" and n. respectively. Pro...
 2.5.53: Suppose that A is a 4 x 4 matrix in the block form. A= [~ g]. where...
Solutions for Chapter 2.5: MATRICES AND LINEAR TRANSFORMATIONS
Full solutions for Elementary Linear Algebra: A Matrix Approach  2nd Edition
ISBN: 9780131871410
Solutions for Chapter 2.5: MATRICES AND LINEAR TRANSFORMATIONS
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra: A Matrix Approach, edition: 2. Elementary Linear Algebra: A Matrix Approach was written by and is associated to the ISBN: 9780131871410. Since 53 problems in chapter 2.5: MATRICES AND LINEAR TRANSFORMATIONS have been answered, more than 22906 students have viewed full stepbystep solutions from this chapter. Chapter 2.5: MATRICES AND LINEAR TRANSFORMATIONS includes 53 full stepbystep solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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

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.)

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Column space C (A) =
space of all combinations of the columns of A.

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.

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.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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.

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.

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

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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

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