 3.1.1: Plot the points , , , , , and on a coordinate plane.
 3.1.2: Plot the points , , , , and on a coordinate plane. Draw the line se...
 3.1.3: Plot the points , , , , and . Describe the set of all points of the...
 3.1.4: Plot the points , , , , and . Describe the set of all points of the...
 3.1.5: Exer. 56: Find the coordinates of the points AF
 3.1.6: Exer. 56: Find the coordinates of the points AF
 3.1.7: Exer. 78: Describe the set of all points in a coordinate plane that...
 3.1.8: Exer. 78: Describe the set of all points in a coordinate plane that...
 3.1.9: Exer. 914: (a) Find the distance between A and B. (b) Find the midp...
 3.1.10: Exer. 914: (a) Find the distance between A and B. (b) Find the midp...
 3.1.11: Exer. 914: (a) Find the distance between A and B. (b) Find the midp...
 3.1.12: Exer. 914: (a) Find the distance between A and B. (b) Find the midp...
 3.1.13: Exer. 914: (a) Find the distance between A and B. (b) Find the midp...
 3.1.14: Exer. 914: (a) Find the distance between A and B. (b) Find the midp...
 3.1.15: Exer. 1516: Show that the triangle with vertices A, B, and C is a r...
 3.1.16: Exer. 1516: Show that the triangle with vertices A, B, and C is a r...
 3.1.17: Show that , , , and are vertices of a square.
 3.1.18: Show that , , , and are vertices of a parallelogram.
 3.1.19: Given , find the coordinates of the point B such that is the midpoi...
 3.1.20: Given and , find the point on segment AB that is threefourths of t...
 3.1.21: Exer. 2122: Prove that C is on the perpendicular bisector of segmen...
 3.1.22: Exer. 2122: Prove that C is on the perpendicular bisector of segmen...
 3.1.23: Exer. 2324: Find a formula that expresses the fact that an arbitrar...
 3.1.24: Exer. 2324: Find a formula that expresses the fact that an arbitrar...
 3.1.25: Find a formula that expresses the fact that is a distance 5 from th...
 3.1.26: Find a formula that states that is a distance from a fixed point . ...
 3.1.27: Find all points on the yaxis that are a distance 6 from
 3.1.28: Find all points on the xaxis that are a distance 5 from
 3.1.29: Find the point with coordinates of the form that is in the third qu...
 3.1.30: Find all points with coordinates of the form that are a distance 3 ...
 3.1.31: For what values of a is the distance between and greater than
 3.1.32: Given and , find a formula not containing radicals that expresses t...
 3.1.33: Prove that the midpoint of the hypotenuse of any right triangle is ...
 3.1.34: Prove that the diagonals of any parallelogram bisect each other. (H...
Solutions for Chapter 3.1: Rectangular Coordinate Systems
Full solutions for Algebra and Trigonometry with Analytic Geometry  12th Edition
ISBN: 9780495559719
Solutions for Chapter 3.1: Rectangular Coordinate Systems
Get Full SolutionsThis textbook survival guide was created for the textbook: Algebra and Trigonometry with Analytic Geometry, edition: 12. This expansive textbook survival guide covers the following chapters and their solutions. Algebra and Trigonometry with Analytic Geometry was written by and is associated to the ISBN: 9780495559719. Chapter 3.1: Rectangular Coordinate Systems includes 34 full stepbystep solutions. Since 34 problems in chapter 3.1: Rectangular Coordinate Systems have been answered, more than 37855 students have viewed full stepbystep solutions from this chapter.

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).

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

GramSchmidt 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.

Iterative method.
A sequence of steps intended to approach the desired solution.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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!

Solvable system Ax = b.
The right side b is in the column space of A.