- 4.1.1: Which of the following relations are functions? For those relations...
- 4.1.2: Give a relation r from to such that(a) r is not a function.(b) r is...
- 4.1.3: Identify the domain, range, and another possible codomain for each ...
- 4.1.4: Assuming that the domain of each of the following functions is the ...
- 4.1.5: (a) Let A be the set and let R be the relation on A given byis prim...
- 4.1.6: Show that the following relations are not functions on .
- 4.1.7: Let the universe be and Sketch the graph of
- 4.1.8: Let U be the universe and with Let be the characteristicfunction of...
- 4.1.9: Give an example of a sequence x such that (a) the range of x is the...
- 4.1.10: For the canonical map find
- 4.1.11: Which of the following are functions from the indicated domain to t...
- 4.1.12: Explain why the functions and are not equal.
- 4.1.13: (a) Prove that the empty set is a function with domain .f(b) Prove ...
- 4.1.14: Complete the proof of Theorem 4.1.1. That is, prove that if (i)and ...
- 4.1.15: Let S be a relation from A to B. We define two projection functions...
- 4.1.16: A metric on a set X is a function such that for all(i)(ii)(iii)(iv)...
- 4.1.17: Suppose that set A has m elements and set B has n elements. We have...
- 4.1.18: (a) Let f be a function from A to B. Define the relation T on A by ...
- 4.1.19: "Assign a grade of A (correct), C (partially correct), or F (failur...
Solutions for Chapter 4.1: Functions as Relations
Full solutions for A Transition to Advanced Mathematics | 7th 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.
peA) = det(A - AI) has peA) = zero matrix.
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.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
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
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.
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.
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.
Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .
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.
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.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
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.
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·