- 6.6.1: Rewrite each expression using only positive exponents. a. 23 b. 52 ...
- 6.6.2: Insert the appropriate symbol (<, =, or >) between each pair of num...
- 6.6.3: Find the exponent of 10 that you need to write each expression in s...
- 6.6.4: The population of a town is currently 45,647. It has been growing a...
- 6.6.5: Juan says that 63 is the same as 63.Write an explanation of how Jua...
- 6.6.6: Use the properties of exponents to rewrite each expression without ...
- 6.6.7: APPLICATION Suppose the annual rate of inflation is about 4%. This ...
- 6.6.8: APPLICATION The population of Japan in 2004 was about 1.3 108. Japa...
- 6.6.9: Decide whether each statement is true or false. Use expanded form t...
- 6.6.10: A large ball of string originally held 1 mile of string. Abigail cu...
- 6.6.11: Suppose 36(1 + 0.5)4 represents the number of bacteria cells in a s...
- 6.6.12: APPLICATION Camila received a $1,200 prize for one of her essays. S...
- 6.6.13: Mini-Investigation In the last few lessons, you have worked with eq...
- 6.6.14: APPLICATION A capacitor is charged with a nine-volt battery. The eq...
- 6.6.15: Set your calculator in scientific notation mode for this problem. a...
Solutions for Chapter 6.6: Zero and Negative Exponents
Full solutions for Discovering Algebra: An Investigative Approach | 2nd Edition
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
peA) = det(A - AI) has peA) = zero matrix.
A = CTC = (L.J]))(L.J]))T for positive definite A.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
Invert A by row operations on [A I] to reach [I A-I].
Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.
A symmetric matrix with eigenvalues of both signs (+ and - ).
Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.
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.
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.
Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
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.
Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.
Solvable system Ax = b.
The right side b is in the column space of A.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).