- 6.4.1: Write each number in scientific notation. a. 34,000,000,000 b. 2,10...
- 6.4.2: Write each number in standard notation. a. 7.4 104 b. 2.134 106 c. ...
- 6.4.3: Use the properties of exponents to rewrite each expression. a. 3x5(...
- 6.4.4: Use the properties of exponents to rewrite each expression. a. 3x2 ...
- 6.4.5: Owen insists on reading his calculators display as three point five...
- 6.4.6: There are approximately 5.58 1021 atoms in a gram of silver. How ma...
- 6.4.7: Because the number of molecules in a given amount of a compound is ...
- 6.4.8: Write each number in scientific notation. How does your calculator ...
- 6.4.9: Cal and Al were assigned this multiplication problem for homework: ...
- 6.4.10: Consider these multiplication expressions: i. (2 105)(3 108) ii. (6...
- 6.4.11: . Americans make almost 2 billion telephone calls each day. (www.br...
- 6.4.12: On average a person sheds 1 million dead skin cells every 40 minute...
- 6.4.13: A light-year is the distance light can travel in one year. This dis...
- 6.4.14: APPLICATION The exponential equation P = 3.8(1 + 0.017)t approximat...
- 6.4.15: Graph y 2(x 5).
Solutions for Chapter 6.4: Scientific Notation for Large Numbers
Full solutions for Discovering Algebra: An Investigative Approach | 2nd Edition
Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
Remove row i and column j; multiply the determinant by (-I)i + j •
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
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.
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).
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.
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
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Invert A by row operations on [A I] to reach [I A-I].
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
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 .
A symmetric matrix with eigenvalues of both signs (+ and - ).
A sequence of steps intended to approach the desired solution.
Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
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.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as A.