 11.1: Rewrite each expression with as few square root symbols as possible.
 11.2: Find the area of the tilted square at right. Use two different stra...
 11.3: Use analytic geometry and deductive reasoning to show that the side...
 11.4: Explain how to draw a square with a side length of units.
 11.5: APPLICATION Is a triangle with side lengths 5 ft, 12 ft, and 13 ft ...
 11.6: Draw this quadrilateral on graph paper. a. Name the coordinates of ...
 11.7: Find the approximate lengths of the legs of this right triangle.
 11.8: You will need a straightedge and a protractor for this exercise. a....
 11.9: A rectangular box has the dimensions shown in the diagram at right....
 11.10: MiniInvestigation If the sides of a triangle are enlarged by a fac...
 11.11: This table gives the normal minimum and maximum January temperature...
 11.12: The HealthyFood Market sells dried fruit by the pound. Jan bought 3...
 11.13: The integers 3 to 16, inclusive, are written on cards and put in a ...
 11.14: Tell whether the relationship between x and y is direct variation, ...
 11.15: Here is a graph of a function f . a. Use words such as linear, nonl...
 11.16: Use the properties of exponents to rewrite each expression. Your an...
 11.17: Here are the running times in minutes of 22 movies in the newrelea...
 11.18: Write the equation for this parabola in a. Factored form. b. Vertex...
 11.19: APPLICATION Six years ago, Mayas grandfather gave her his baseball ...
 11.20: If f (x) = x2 +  x  4, find a. f (5) b. f (2) c. f (7) f (4) d. f...
 11.21: APPLICATION The Galaxy of Shoes store is having a 22nd anniversary ...
 11.22: The image of the black rectangle after a transformation is shown in...
 11.23: Solve for x. a. 0 = (x + 5)(x 2) b. 0 = x2 + 8x + 16 c. x2 5x = 2x ...
 11.24: Give the equation for each graph, written as transformations of a p...
 11.25: APPLICATION Zoe Kovalesky visits companies and teaches the employee...
 11.26: The vertices of ABC are A(6, 1), B(2, 7), and C(10, 1). Do 26ad bef...
Solutions for Chapter 11: Introduction to Geometry
Full solutions for Discovering Algebra: An Investigative Approach  2nd Edition
ISBN: 9781559537636
Solutions for Chapter 11: Introduction to Geometry
Get Full SolutionsDiscovering Algebra: An Investigative Approach was written by and is associated to the ISBN: 9781559537636. This textbook survival guide was created for the textbook: Discovering Algebra: An Investigative Approach, edition: 2. Chapter 11: Introduction to Geometry includes 26 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 26 problems in chapter 11: Introduction to Geometry have been answered, more than 9690 students have viewed full stepbystep solutions from this chapter.

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

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

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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 IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

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.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B IIĀ·

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

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