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# 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

Solutions for Chapter 11
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##### ISBN: 9781559537636

Discovering 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 step-by-step 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 step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
• 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 IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad

• Identity matrix I (or In).

Diagonal entries = 1, off-diagonal entries = 0.

• Incidence matrix of a directed graph.

The m by n edge-node 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.

• 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.

• 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.

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