 4.1.1: Find the slope of each line using a slope triangle or the slope for...
 4.1.2: Find the slope of the line through each pair of points. Then name a...
 4.1.3: Given the slope of a line and one point on the line, name two other...
 4.1.4: Run the LINES program five times. Start by playing the easy level o...
 4.1.5: Each table gives the coordinates of four points on a different line...
 4.1.6: Consider lines a and b shown in the graph at right. a. How are the ...
 4.1.7: APPLICATION Recall Hectors Internet use from the investigation. You...
 4.1.8: If a and c are the lengths of the vertical and horizontal segments ...
 4.1.9: This line has a slope of 1. Graph it on your own paper. a. Draw a s...
 4.1.10: APPLICATION A hotair balloonist gathered the data in this table. a...
 4.1.11: When you make a scatter plot of realworld data, you may see a line...
 4.1.12: The base of a triangle was recorded as 18.3 0.1 cm and the height w...
 4.1.13: Calista has five brothers. The mean of her brothers ages is 10 year...
 4.1.14: Enter {3, 1, 2, 8, 10} into list L1 on your calculator. a. Write a ...
 4.1.15: Convert each decimal number to a percent. a. 0.85 b. 1.50 c. 0.065 ...
 4.1.16: The equation 7x 10 = 2x + 3 is solved by balancing. Explain what ha...
Solutions for Chapter 4.1: A Formula for Slope
Full solutions for Discovering Algebra: An Investigative Approach  2nd Edition
ISBN: 9781559537636
Solutions for Chapter 4.1: A Formula for Slope
Get Full SolutionsThis textbook survival guide was created for the textbook: Discovering Algebra: An Investigative Approach, edition: 2. This expansive textbook survival guide covers the following chapters and their solutions. Since 16 problems in chapter 4.1: A Formula for Slope have been answered, more than 4577 students have viewed full stepbystep solutions from this chapter. Chapter 4.1: A Formula for Slope includes 16 full stepbystep solutions. Discovering Algebra: An Investigative Approach was written by and is associated to the ISBN: 9781559537636.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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).

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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

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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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 ().

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

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.