- 1-2.1-2.1: In Exercises 13, perform the indicated operation or operations 8 (1...
- 1-2.1-2.2: In Exercises 13, perform the indicated operation or operations (3)(...
- 1-2.1-2.3: In Exercises 13, perform the indicated operation or operations (8 1...
- 1-2.1-2.4: Simplify: 2 5x 3(x 7).
- 1-2.1-2.5: List all the rational numbers in this set: 4, 1 3 , 0, 2, 4, p 2 , ...
- 1-2.1-2.6: Write as an algebraic expression, using x to represent the number: ...
- 1-2.1-2.7: Insert either or in the shaded area to make a true statement: 10,00...
- 1-2.1-2.8: 6(4x 1 5y).
- 1-2.1-2.9: Does the formula underestimate or overestimate the percentage of se...
- 1-2.1-2.10: If trends shown by the formula continue, when will only 33% of high...
- 1-2.1-2.11: In Exercises 1112, solve each equation. 5 6(x 2) x 14
- 1-2.1-2.12: In Exercises 1112, solve each equation. x 5 2 x 3
- 1-2.1-2.13: Solve for A: V 1 3 Ah.
- 1-2.1-2.14: 48 is 30% of what number?
- 1-2.1-2.15: The length of a rectangular parking lot is 10 yards less than twice...
- 1-2.1-2.16: A gas station owner makes a profit of 40 cents per gallon of gasoli...
- 1-2.1-2.17: Express the solution set of x 12 in interval notation and graph the...
- 1-2.1-2.18: 3 3x 12
- 1-2.1-2.19: 5 2(3 x) 2(2x 5) 1
- 1-2.1-2.20: You take a summer job selling medical supplies. You are paid $600 p...
Solutions for Chapter 1-2: Cumulative Review Exercises
Full solutions for Introductory & Intermediate Algebra for College Students | 4th Edition
Tv = Av + Vo = linear transformation plus shift.
Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
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.)
Characteristic equation det(A - AI) = O.
The n roots are the eigenvalues of A.
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 sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
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.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
A directed graph that has constants Cl, ... , Cm associated with the edges.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Every v in V is orthogonal to every w in W.
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
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 ().
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.