Question

Let A =

v1 | v2 | v3 | |

v1 | 2 | 8 | 16 |

v2 | 8 | 0 | 4 |

v3 | 16 | 4 | 1 |

be the ADJACENCY MATRIX for an undirected graph G. Solve the following:

1) Determine the number of edges of G

2) Determine the total degree of G

3) Determine the degree of each vertex of G

4) Determine the number of different walks of length 2 from vertex v3 to v1

5) Does G have an Euler circuit? Explain

Answer #1

1) Number of edges in G= no. of loops + number of edges between two different vertices

Number of loops = 2+1 =3

Number of edges between vertices = #edges between v1-v2 + #edges between v1-v3 + #edges between v2-v3 = 8+16+4=28

Thus number of edges in G = 31

2) Total degree of G = sum of degree of each vertices = degree of v1 + degree of v2 + degree of v3 = 26 + 12 + 21 = 59

3) Degree of each vertex = number of edges incident from each vertex. Thus:

degree of v1 = 26

degree of v2 = 12

degree of v3 = 21

4) Number of different walks of length from v3 to v1:

case(i) v3-v1-v1 = number of walks = 16*2 = 32

case(ii) v3-v3-v1 = 1*16 =16

case(iii) v3-v2-v1 = 4*8 = 32

thus total number of different walks = 80

5) Yes, the graph is Eulerian as each vertex is connected to other vertices.

Let H=Span{v1,v2} and
K=Span{v3,v4}, where
v1,v2,v3,v4 are given
below.
v1 = [3 2 5], v2 =[4 2 6], v3
=[5 -1 1], v4 =[0 -21 -9]
Then H and K are subspaces of R3 . In fact, H and K
are planes in R3 through the origin, and they intersect
in a line through 0. Find a nonzero vector w that
generates that line.
w = { _______ }

Let S = {v1, v2, v3, v4} be a given basis of R ^4 . Suppose that
A is a (3 × 4) matrix with the following properties: Av1 = 0, A(v1
+ 2v4) = 0, Av2 =[ 1 1 1 ] T , Av3 = [ 0 −1 −4
]T . Find a basis for N (A), and a basis for R(A). Fully
justify your answer.

5. Let v1 = (1/3,−2/3,2/3), v2 = (2/3,−1/3,−2/3) and v3 =
(2/3,2/3,1/3).
(a) Verify that v1, v2, v3 is an orthonormal basis of R 3 .
(b) Determine the coordinates of x = (9, 10, 11), v1 − 4v2 and
v3 with respect to v1, v2, v3.

Let v1=(0,1,2,3),v2=(1,0,-1,0),v3=(0,4,-1,2), and v4=(0,5,1,5).
Let S=(v1,v2,v3,v4)
(1)find a basis for span(S)
(2)is the vector e1=(1,0,0,0) in the span of S? Why?

let v1=[1,0,10], v2=[0,1,0,1] and let W be the
subspace of R^4 spanned by v1 and v2.
A. convert {v1,v2} into an orhonormal basis of W.
Basis =
B.find the projection of b=[-1,-2,-2,-1] onto W
C.find two linear independent vectors in R^4
perpendicular to W.
vectors =

Determine if the vectors v1= (3, 0, -3, 6),
v2 = ( 0, 2, 3, 1), and v3 = (0, -2, 2, 0 )
form a linearly dependent set in R 4. Is it a basis of
R4 ?

Let X be a real vector space. Suppose {⃗v1,⃗v2,⃗v3} ⊂ X is a
linearly independent set, and suppose {w⃗1,w⃗2,w⃗3} ⊂ X is a
linearly dependent set. Define V = span{⃗v1,⃗v2,⃗v3} and W =
span{w⃗1,w⃗2,w⃗3}.
(a) Is there a linear transformation P : V → W such that P(⃗vi)
= w⃗i for i = 1, 2, 3?
(b) Is there a linear transformation Q : W → V such that Q(w⃗i)
= ⃗vi for i = 1, 2, 3?
Hint: the...

Consider four vectors v1 = [1,1,1,1], v2 = [-1,0,1,2], v3 =
[a,1,0,b], and v4 = [3,2,a+b,0], where a and b are parameters. Find
all conditions on the values of a and b (if any) for which:
1. The number of linearly independent vectors in this collection
is 1.
2. The number of linearly independent vectors in this collection
is 2.
3. The number of linearly independent vectors in this collection
is 3.
4. The number of linearly independent vectors in...

Determine all real numbers a for which the vectors
v1 = (1,−1,1,a,2)
v2 = (−1,0,0,1,0)
v3 = (1,2,a + 1,1,0)
v4 = (2,0,a + 3,2a + 3,4)
make a linearly independent set. For which values of a does the
set contain at least three linearly independent vectors?

Let G be the graph having 3 vertices A, B and C. There are five
edges connecting A and B and three edges connecting B and C. Solve
the following:
1) Determine the number of paths from A to B
2) Determine the number of different trails of length 6 from A
to B

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 11 minutes ago

asked 22 minutes ago

asked 23 minutes ago

asked 23 minutes ago

asked 25 minutes ago

asked 49 minutes ago

asked 51 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago