 9.4.9.4.1: Fill in the blanks. The procedure used to eliminate the xyterm in ...
 9.4.9.4.2: Fill in the blanks. Quantities that are equal in both the original ...
 9.4.9.4.3: Fill in the blanks. The quantity is called the _______ of the equat...
 9.4.9.4.4: In Exercises 314, rotate the axes to eliminate the xyterm in the e...
 9.4.9.4.5: In Exercises 314, rotate the axes to eliminate the xyterm in the e...
 9.4.9.4.6: In Exercises 314, rotate the axes to eliminate the xyterm in the e...
 9.4.9.4.7: In Exercises 314, rotate the axes to eliminate the xyterm in the e...
 9.4.9.4.8: In Exercises 314, rotate the axes to eliminate the xyterm in the e...
 9.4.9.4.9: In Exercises 314, rotate the axes to eliminate the xyterm in the e...
 9.4.9.4.10: In Exercises 314, rotate the axes to eliminate the xyterm in the e...
 9.4.9.4.11: In Exercises 314, rotate the axes to eliminate the xyterm in the e...
 9.4.9.4.12: In Exercises 314, rotate the axes to eliminate the xyterm in the e...
 9.4.9.4.13: In Exercises 314, rotate the axes to eliminate the xyterm in the e...
 9.4.9.4.14: In Exercises 314, rotate the axes to eliminate the xyterm in the e...
 9.4.9.4.15: In Exercises 1520, use a graphing utility to graph the conic. Deter...
 9.4.9.4.16: In Exercises 1520, use a graphing utility to graph the conic. Deter...
 9.4.9.4.17: In Exercises 1520, use a graphing utility to graph the conic. Deter...
 9.4.9.4.18: In Exercises 1520, use a graphing utility to graph the conic. Deter...
 9.4.9.4.19: In Exercises 1520, use a graphing utility to graph the conic. Deter...
 9.4.9.4.20: In Exercises 1520, use a graphing utility to graph the conic. Deter...
 9.4.9.4.21: In Exercises 2126, match the graph with its equation. [The graphs a...
 9.4.9.4.22: In Exercises 2126, match the graph with its equation. [The graphs a...
 9.4.9.4.23: In Exercises 2126, match the graph with its equation. [The graphs a...
 9.4.9.4.24: In Exercises 2126, match the graph with its equation. [The graphs a...
 9.4.9.4.25: In Exercises 2126, match the graph with its equation. [The graphs a...
 9.4.9.4.26: In Exercises 2126, match the graph with its equation. [The graphs a...
 9.4.9.4.27: In Exercises 2734, (a) use the discriminant to classify the graph o...
 9.4.9.4.28: In Exercises 2734, (a) use the discriminant to classify the graph o...
 9.4.9.4.29: In Exercises 2734, (a) use the discriminant to classify the graph o...
 9.4.9.4.30: In Exercises 2734, (a) use the discriminant to classify the graph o...
 9.4.9.4.31: In Exercises 2734, (a) use the discriminant to classify the graph o...
 9.4.9.4.32: In Exercises 2734, (a) use the discriminant to classify the graph o...
 9.4.9.4.33: In Exercises 2734, (a) use the discriminant to classify the graph o...
 9.4.9.4.34: In Exercises 2734, (a) use the discriminant to classify the graph o...
 9.4.9.4.35: In Exercises 3538, sketch (if possible) the graph of the degenerate...
 9.4.9.4.36: In Exercises 3538, sketch (if possible) the graph of the degenerate...
 9.4.9.4.37: In Exercises 3538, sketch (if possible) the graph of the degenerate...
 9.4.9.4.38: In Exercises 3538, sketch (if possible) the graph of the degenerate...
 9.4.9.4.39: In Exercises 3946, solve the system of quadratic equations algebrai...
 9.4.9.4.40: In Exercises 3946, solve the system of quadratic equations algebrai...
 9.4.9.4.41: In Exercises 3946, solve the system of quadratic equations algebrai...
 9.4.9.4.42: In Exercises 3946, solve the system of quadratic equations algebrai...
 9.4.9.4.43: In Exercises 3946, solve the system of quadratic equations algebrai...
 9.4.9.4.44: In Exercises 3946, solve the system of quadratic equations algebrai...
 9.4.9.4.45: In Exercises 3946, solve the system of quadratic equations algebrai...
 9.4.9.4.46: In Exercises 3946, solve the system of quadratic equations algebrai...
 9.4.9.4.47: In Exercises 4752, solve the system of quadratic equations algebrai...
 9.4.9.4.48: In Exercises 4752, solve the system of quadratic equations algebrai...
 9.4.9.4.49: In Exercises 4752, solve the system of quadratic equations algebrai...
 9.4.9.4.50: In Exercises 4752, solve the system of quadratic equations algebrai...
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 9.4.9.4.53: True or False? In Exercises 53 and 54, determine whether the statem...
 9.4.9.4.54: True or False? In Exercises 53 and 54, determine whether the statem...
 9.4.9.4.55: In Exercises 5558, sketch the graph of the rational function. Ident...
 9.4.9.4.56: In Exercises 5558, sketch the graph of the rational function. Ident...
 9.4.9.4.57: In Exercises 5558, sketch the graph of the rational function. Ident...
 9.4.9.4.58: In Exercises 5558, sketch the graph of the rational function. Ident...
 9.4.9.4.59: In Exercises 5962, if possible, find (a) AB, (b) BA, and (c)
 9.4.9.4.60: In Exercises 5962, if possible, find (a) AB, (b) BA, and (c)
 9.4.9.4.61: In Exercises 5962, if possible, find (a) AB, (b) BA, and (c)
 9.4.9.4.62: In Exercises 5962, if possible, find (a) AB, (b) BA, and (c)
 9.4.9.4.63: In Exercises 6370, graph the function.
 9.4.9.4.64: In Exercises 6370, graph the function.
 9.4.9.4.65: In Exercises 6370, graph the function.
 9.4.9.4.66: In Exercises 6370, graph the function.
 9.4.9.4.67: In Exercises 6370, graph the function.
 9.4.9.4.68: In Exercises 6370, graph the function.
 9.4.9.4.69: In Exercises 6370, graph the function.
 9.4.9.4.70: In Exercises 6370, graph the function.
 9.4.9.4.71: In Exercises 7174, find the area of the triangle.
 9.4.9.4.72: In Exercises 7174, find the area of the triangle.
 9.4.9.4.73: In Exercises 7174, find the area of the triangle.
 9.4.9.4.74: In Exercises 7174, find the area of the triangle.
Solutions for Chapter 9.4: Rotation and Systems of Quadratic Equations
Full solutions for Precalculus With Limits A Graphing Approach  5th Edition
ISBN: 9780618851522
Solutions for Chapter 9.4: Rotation and Systems of Quadratic Equations
Get Full SolutionsChapter 9.4: Rotation and Systems of Quadratic Equations includes 74 full stepbystep solutions. This textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5. Since 74 problems in chapter 9.4: Rotation and Systems of Quadratic Equations have been answered, more than 33079 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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

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

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

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.

Outer product uv T
= column times row = rank one matrix.

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

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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).

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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.