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 1.81: Banking A bank lends out $9000 at 7% simple interest. At the end of...
 1.82: Financial Planning Steve, a recent retiree, requires $5000 per year...
 1.83: Lightning and Thunder A flash of lightning is seen, and the resulti...
 1.84: Physics: Intensity of Light The intensity I (in candlepower) of a c...
 1.85: Extent of Search and Rescue A search plane has a cruising speed of ...
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 1.93: Chemistry: Salt Solutions How much water should be added to 64 ounc...
 1.94: Chemistry: Salt Solutions How much water must be evaporated from 64...
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 1.99: Using Two Pumps An 8horsepower (hp) pump can fill a tank in 8 hour...
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 1.101: Finance An inheritance of $900,000 is to be divided among Scott, Al...
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Solutions for Chapter 1: Equations and Inequalities
Full solutions for College Algebra  9th Edition
ISBN: 9780321716811
Solutions for Chapter 1: Equations and Inequalities
Get Full SolutionsThis textbook survival guide was created for the textbook: College Algebra, edition: 9. Since 105 problems in chapter 1: Equations and Inequalities have been answered, more than 15336 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. College Algebra was written by and is associated to the ISBN: 9780321716811. Chapter 1: Equations and Inequalities includes 105 full stepbystep solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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.

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.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Stiffness matrix
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.

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

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.