- 8.2.1: Use f (x) = 2 | x + 4 | + 1 to find a. f (5) b. f (6) c. f (2) + 3 ...
- 8.2.2: List L1 and list L2 contain coordinates for three points on the gra...
- 8.2.3: Give the coordinates of the vertex for each graph.
- 8.2.4: Use a calculator to graph each equation. Describe the graph as a tr...
- 8.2.5: Write an equation for each of these transformations. a. Translate t...
- 8.2.6: Describe each graph in Exercise 3 as a transformation of y = | x | ...
- 8.2.7: This graph shows Beths distance from her teacher as she turns in he...
- 8.2.8: Graph Y1(x) = abs(x) on your calculator. Predict what each graph wi...
- 8.2.9: Describe how the graph of y = x2 will be transformed if you replace...
- 8.2.10: APPLICATION The equation y = a bx models the decreasing voltage of ...
- 8.2.11: Mini-Investigation Recall that an exponential equation in the form ...
- 8.2.12: APPLICATION In 2004, the world population was estimated to be 6.4 b...
- 8.2.13: The graph of a linear equation of the form y = bx passes through (0...
- 8.2.14: Drews teacher gives skill-building quizzes at the start of each cla...
- 8.2.15: Solve each system of equations.
Solutions for Chapter 8.2: Translating Graphs
Full solutions for Discovering Algebra: An Investigative Approach | 2nd Edition
Upper triangular systems are solved in reverse order Xn to Xl.
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!
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.
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).
cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
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
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 Fn-1c can be computed with ne/2 multiplications. Revolutionary.
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.
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
Length II x II.
Square root of x T x (Pythagoras in n dimensions).
A directed graph that has constants Cl, ... , Cm associated with the edges.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Every v in V is orthogonal to every w in W.
Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= 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.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.
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