 11.1  11.3.11.111.311.4: In Exercises 113, solve each equation by the method of your choice....
 11.1  11.3.11.111.311.5: In Exercises 113, solve each equation by the method of your choice....
 11.1  11.3.11.111.311.6: In Exercises 113, solve each equation by the method of your choice....
 11.1  11.3.11.111.311.7: In Exercises 113, solve each equation by the method of your choice....
 11.1  11.3.11.111.311.8: In Exercises 113, solve each equation by the method of your choice....
 11.1  11.3.11.111.311.9: In Exercises 113, solve each equation by the method of your choice....
 11.1  11.3.11.111.311.10: In Exercises 113, solve each equation by the method of your choice....
 11.1  11.3.11.111.311.11: In Exercises 113, solve each equation by the method of your choice....
 11.1  11.3.11.111.311.12: In Exercises 113, solve each equation by the method of your choice....
 11.1  11.3.11.111.311.13: In Exercises 113, solve each equation by the method of your choice....
 11.1  11.3.11.111.311.14: In Exercises 113, solve each equation by the method of your choice....
 11.1  11.3.11.111.311.15: In Exercises 113, solve each equation by the method of your choice....
 11.1  11.3.11.111.311.16: In Exercises 113, solve each equation by the method of your choice....
 11.1  11.3.11.111.311.17: Solve by completing the square: x2 10x 3 0.
 11.1  11.3.11.111.311.18: (2, 2) and (2, 2)
 11.1  11.3.11.111.311.19: (5, 8) and (10, 14)
 11.1  11.3.11.111.311.20: In Exercises 1720, graph the given quadratic function. Give each fu...
 11.1  11.3.11.111.311.21: In Exercises 1720, graph the given quadratic function. Give each fu...
 11.1  11.3.11.111.311.22: In Exercises 1720, graph the given quadratic function. Give each fu...
 11.1  11.3.11.111.311.23: In Exercises 1720, graph the given quadratic function. Give each fu...
 11.1  11.3.11.111.311.24: In Exercises 2122, without solving the equation, determine the numb...
 11.1  11.3.11.111.311.25: In Exercises 2122, without solving the equation, determine the numb...
 11.1  11.3.11.111.311.26: In Exercises 2324, write a quadratic equation in standard form with...
 11.1  11.3.11.111.311.27: In Exercises 2324, write a quadratic equation in standard form with...
 11.1  11.3.11.111.311.28: A company manufactures and sells bath cabinets. The function P(x) x...
 11.1  11.3.11.111.311.29: Among all pairs of numbers whose sum is 18, find a pair whose produ...
 11.1  11.3.11.111.311.30: The base of a triangle measures 40 inches minus twice the measure o...
Solutions for Chapter 11.1  11.3: MIDCHAPTER CHECK POINT
Full solutions for Introductory & Intermediate Algebra for College Students  4th Edition
ISBN: 9780321758941
Solutions for Chapter 11.1  11.3: MIDCHAPTER CHECK POINT
Get Full SolutionsChapter 11.1  11.3: MIDCHAPTER CHECK POINT includes 27 full stepbystep solutions. This textbook survival guide was created for the textbook: Introductory & Intermediate Algebra for College Students, edition: 4. Since 27 problems in chapter 11.1  11.3: MIDCHAPTER CHECK POINT have been answered, more than 71456 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Introductory & Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758941.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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

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

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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

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

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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.

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

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

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

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

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