Answered: 1.2.15. (!) Let W be a closed walk of… | bartleby

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.5: Graphs Of Functions

Problem 47E

See similar textbooks

Related questions

Question

1.2.15. (!) Let W be a closed walk of length at least 1 that does not contain a cycle.
Prove that some edge of W repeats immediately (once in each direction).

Expert Solution

Step by step

Solved in 2 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Blurred answer

Similar questions

Determine whether the following sets are linearly dependent or linearlyindependent. (a) {(; )( )} . -9} in Maxa(R) 4 (b) {(- ).(- 9G -)} in M2x2(R) 4 (c) {x 3 + 2x2, -x2 + 3x + 1, x3 – x2 + 2x – 1} in P3(R) (d) {x 3 – x, 2x2 + 4, -2x3 + 3x2 + 2x + 6} in P3(R) (e) {(1, -1, 2), (1, -2, 1), (1, 1, 4)} in R3 (f) {(1, –1, 2), (2, 0, 1), (–1, 2, –1)} in R3 {(÷ ) ( -) (; 0. 2 ).(- )} in Max2(R) (g) (m) {(-; ') (? ). -2)} in M2x2(R) () (x 4 - х3 + 5x2 - 8х + 6, -х4 + х3 - 5х2 + 5х - 3, x 4 + 3x2 – 3x + 5, 2x4 + 3x3 + 4x2 – x + 1, x3 – x + 2} in P4(R) ) (x4 - х3 + 5x2 - 8х + 6, -х4 + x3 - 5x2 + 5х - 3, х4 + 3x2 - 3х + 5, 2х4 + х3 + 4x2 + 8х} in P4(R)
). 4. Rectangle DEFG with vertices D(-4, 3), E(O, 2), F(-2, -6), and G(-6, -5): (x, y) (x+ 4, y + 1) D'( E'( . F' G' - onent form. 7. OGina Wison (AI Things Algebra, LLC). 2015-2018
3. [10 marks] Let Go = (V,E) and G₁ = (V,E₁) be two graphs on the same set of vertices. Let (V, EU E1), so that (u, v) is an edge of H if and only if (u, v) is an edge of Go or of G1 (or of both). H = (a) Show that if Go and G₁ are both Eulerian and En E₁ = Ø (i.e., Go and G₁ have no edges in common), then H is also Eulerian. (b) Give an example where Go and G₁ are both Eulerian, but H is not Eulerian.
Prove: Let F,F'be forests on the same set of vertices, with |E(F)|< |E(F')|. Show that F'has an edge e such that F + e is again a forest.
6. The covariance between X and Y is calculated and recorded as SXY (and sxy is a negative number). A linear transformation is performed on X. The linearly transformed variable is called W, where W; = 5 + Xi. The covariance between W and Y is calculated and recorded as sWY. Which statement is correct? A) В) In this example, swy would be less than sxY. In this example, swY Would be equal to sxY. In this example, swy would be greater than SXXY. 7. The covariance between X and Y is calculated and recorded as sxy (and sxy is a positive number). A linear transformation is performed on X. The linearly transformed variable is called W, where W; = (4)X;. The covariance between W and Y is calculated and recorded as sWY. Which statement is correct? In this example, sxY would be less than swy. In this example, sxY Would be equal to swy. In this example, sxY would be greater than sWY. A) B) 8. The correlation between X and Y is calculated and recorded as rxY (and rxy is positive). A linear…
18.
8.2 Affine Independence 477 14. (T/F) Given S = {b1.....bx} in R", each p in aff S has a unique represcntation as an affine combination of bi...bk- %3D 15. (T/F) If S = {v.....v} is an affinely independent set in R" and if p in R" has a negative barycentric coordinate determincd by S, then p is not in aff S. 16. (T/F) When color information is specificd at cach vertex V1, V2. V3 of a triangle in R', then the color may be inter- polated at a point p in aff {v, V2, V3} using the barycentric coordinates of p. 17. (T/F) If VỊ, V2, V3. a, and b are in R3 and if a ray a +ib for 120 intersects the triangle with vertices v. V2, and v3, then the barycentric coordinates of the interscction point arc all nonnegative. pue 18. (T/F) If T is a triangle in R and if a point p is on an edge of the triangle, then the barycentric coordinates of p (for this triangle) arc not all positive
7. [10 marks] Let G = (V,E) be a 3-connected graph. We prove that for every x, y, z Є V, there is a cycle in G on which x, y, and z all lie. (a) First prove that there are two internally disjoint xy-paths Po and P₁. (b) If z is on either Po or P₁, then combining Po and P₁ produces a cycle on which x, y, and z all lie. So assume that z is not on Po and not on P₁. Now prove that there are three paths Qo, Q1, and Q2 such that: ⚫each Qi starts at z; • each Qi ends at a vertex w; that is on Po or on P₁, where wo, w₁, and w₂ are distinct; the paths Qo, Q1, Q2 are disjoint from each other (except at the start vertex 2) and are disjoint from the paths Po and P₁ (except at the end vertices wo, W1, and w₂). (c) Use paths Po, P₁, Qo, Q1, and Q2 to prove that there is a cycle on which x, y, and z all lie. (To do this, notice that two of the w; must be on the same Pj.)
3.1.10. Let M and N be matchings in a graph G, with |M| > |N|. Prove that there exist matchings M' and N' in G such that |M'| = |M| – 1, |N'| = |N|+ 1, and M', N' have the same union and intersection (as edge sets) as M, N.
For each of the following, determine (with argument) whether the given function T is linear. 1. The map T : Mat2,2(R) → Mat2,2(R) given by r(: ) - (**" .-) a b T а +с а — d d 2. The map T : Mat2,2(R) → Mat2,2(R) given by a + c 1 a T с а 1 а — d 3. The map T : Matn.n(R) → Mat,n.n(R) given by T(A) = A · B – B · A where B E Mat,.n (R) is fixed. 3,n 4. The map T : Mat,.n (R) → R given by T(A) = Tr(A · B – B · A) where Be Mat„,n(IR) is fixed. 5. The map T : Fun(R, R) → Fun(R, R) given by T(f(t)) = f(t²). 6. The map T : Fun(R, R) → Fun(IR, IR) given by T(F (t)) = (f(t))².
2.4 #8

Recommended textbooks for you

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Elementary Linear Algebra (MindTap Course List)

Elementary Linear Algebra (MindTap Course List)

Algebra

ISBN:

9781305658004

Author:

Ron Larson

Publisher:

Cengage Learning

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

Algebra

ISBN:

9781285463247

Author:

David Poole

Publisher:

Cengage Learning

Elementary Linear Algebra (MindTap Course List)

Elementary Linear Algebra (MindTap Course List)

Algebra

ISBN:

9781305658004

Author:

Ron Larson

Publisher:

Cengage Learning

SEE MORE TEXTBOOKS