- 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
Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).
Tv = Av + Vo = linear transformation plus shift.
Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)
A = CTC = (L.J]))(L.J]))T for positive definite A.
Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Gram-Schmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
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.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Symmetric matrix A.
The transpose is AT = A, and aU = a ji. A-I is also symmetric.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.