- 11.1.1: Find the slope of each line.a. y = 0.8(x 4) + 7 b. y = 5 2x c. y = ...
- 11.1.2: Determine whether each pair of lines is parallel, perpendicular, or...
- 11.1.3: Line has slope 1.2. What is the slope of line p that is parallel to...
- 11.1.4: Line has slope 1.2. Line m is perpendicular to line . a. What is th...
- 11.1.5: Find the equation in point-slope form of the line that passes throu...
- 11.1.6: Name each quadrilateral using the most specific term that describes...
- 11.1.7: For Exercises 714, plot each set of points on graph paper and conne...
- 11.1.8: For Exercises 714, plot each set of points on graph paper and conne...
- 11.1.9: For Exercises 714, plot each set of points on graph paper and conne...
- 11.1.10: For Exercises 714, plot each set of points on graph paper and conne...
- 11.1.11: For Exercises 714, plot each set of points on graph paper and conne...
- 11.1.12: For Exercises 714, plot each set of points on graph paper and conne...
- 11.1.13: For Exercises 714, plot each set of points on graph paper and conne...
- 11.1.14: For Exercises 714, plot each set of points on graph paper and conne...
- 11.1.15: Al says you can define a right trapezoid as a quadrilateral with ex...
- 11.1.16: For each situation, identify whether inductive or deductive reasoni...
- 11.1.17: Multiply and combine like terms. a. x(x + 2)(2x 1) b. (0.1x 2.1)(0....
- 11.1.18: Find the value halfway between a. 3 and 11 b. 4 and 7 c. 12 and 1 d...
Solutions for Chapter 11.1: Parallel and Perpendicular
Full solutions for Discovering Algebra: An Investigative Approach | 2nd Edition
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.
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).
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n - r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.