- 2.8.1: Evaluate each expression without a calculator. Then check your resu...
- 2.8.2: The equation = C can be used to change temperatures in Fahrenheit t...
- 2.8.3: Evaluate each expression if x = 6. a. 2x + 3 b. 2(x + 3) c. 5x 13 d.
- 2.8.4: For each equation identify the order of operations. Then work backw...
- 2.8.5: To change from miles per hour to feet per second, you can multiply ...
- 2.8.6: Justine asked her group members to do this calculation: Pick a numb...
- 2.8.7: The final answer to the sequence of calculations shown at right is ...
- 2.8.8: The sequence of operations at right will always give you a differen...
- 2.8.9: Consider the expression a. Find the value of the expression if x = ...
- 2.8.10: Solve each equation with an undo table.a. 3(x 5) + 8 = 14.8 b. c. d.
- 2.8.11: Consider the expression a. Find the value of the expression if x = ...
- 2.8.12: The equation D = 6 + 0.4(t 5) represents the depth of water, in inc...
- 2.8.13: Find the errors in this undo table and correct them.
- 2.8.14: An electric slot car travels at a scale speed of 200 mi/h, meaning ...
- 2.8.15: Find a rate for each situation. Then use the rate to answer the que...
- 2.8.16: Natalie works in a shop that sells mixed nuts. Alice drops by and d...
Solutions for Chapter 2.8: Undoing Operations
Full solutions for Discovering Algebra: An Investigative Approach | 2nd Edition
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.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
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.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Invert A by row operations on [A I] to reach [I A-I].
Gram-Schmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.
Hilbert matrix hilb(n).
Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.
Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
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.
Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!
Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.