Being students of business and economics, you should be familiar with the concepts of capitalism and socialism, or generally speaking, command and market-based economies. What would like you to do is list 2 benefits and 2 drawbacks to each system, and then briefly explain why examples of pure socialism and pure capitalism tend to be more theoretical in nature and less likely to be found in reality. In order to receive full credit on this discussion board, you will also need to comment (respectfully) on at least one other student's post.

In this program you will read a file specifying a directed graph G as a list of edges. Each edge u →v is listed on one line as u v. The input file simply lists the edges in arbitrary order as pairs of vertices, with each edge on a separate line. The vertices are numbered in order from 1 to the total number of vertices. The program outputs the out-degree sequence for G^SCC in increasing order.

For example for the input below:

2 1

3 2

1 3

1 5

3 4

4 5

6 7

5 6

7 4

3 11

2 8

8 9

10 8

9 10

9 4

9 7

10 11

7 11

6 11

the corresponding output is:

0 1 2 3

You should be able to implement this program to run in linear time (time proportional to the total number of vertices plus the total number of edges in the input graph)

Use Kosaraju's Algorithm to find the SCC

Please code this in C

Show that if V is finite-dimensional and W is infinite-dimensional, then V and W are NOT isomorphic.

Let U be a subspace of V . Prove that dim U ⊥ = dim V −dim U.

I have a question in that if v is s any nonzero vector, and v is positioned with its initial point at the origin, then the terminal points of all scalar multiples of v will occur at all the points on a straight line through the origin. But if we want to find two vectors, let's say m and n, that are parallel to each other, we need to determine whether they are multiples of each other.

So my question here is: what's the difference between scalar multiples and multiples? In another word, vectors within the same line are parallel to each other because this would be a special form of parallel?

Questionnnnnnn

a. Let V and W be vector spaces and T : V → W a linear transformation. If {T(v1), . . . T(vn)} is linearly independent in W, show that {v1, . . . vn} is linearly independent in V .

b. Define similar matrices

c Let A1, A2 and A3 be n × n matrices. Show that if A1 is similar to A2 and A2 is similar to A3, then A1 is similar to A3.

d. Show that similar matrices have the same characteristic polynomial and eigenvalues.

e. Determine whether the following mappings are linear transformations.

T : V → R defined by T(x) = hx, vi, where v is a fixed nonzero vector in the real inner product space V .

The linear transformation is such that for any v in R², T(v) = Av.

a) Use this relation to find the image of the vectors v₁ = [-3,2]^T and v₂ = [2,3]^T. For the following transformations take k = 0.5 first then k = 3,

T₁(x,y) = (kx,y)

T₂(x,y) = (x,ky)

T₃(x,y) = (x+ky,y)

T₄(x,y) = (x,kx+y)

For T₅ take theta = (pi/4) and then theta = (pi/2)

T₅(x,y) = (cos(theta)x - sin(theta)y, sin(theta)x + cos(theta)y)

b) Plot v₁ and v₂ and their images under the transformations. Write a short description saying what the transformations is doing to the vectors.

Let V be a vector space and let U and W be subspaces of V . Show that the sum U + W = {u + w : u ∈ U and w ∈ W} is a subspace of V .

Suppose u, and v are vectors in R m, such that ∥u∥ = 1, ∥v∥ = 4, ∥u + v∥ = 5. Find the inner product 〈u, v〉.

Suppose {a1, · · · ak} are orthonormal vectors in R m. Show that {a1, · · · ak} is a linearly independent set.

2) Let v, w, and x be vectors in Rn.
a) If v is the zero vector, what geometric object represents all linear
combinations of v?
b) Same question as a), except now for a nonzero v.
c) Same question as a) except now for nonzero vectors v and w (be care-
ful!).
d) Same question as a) except now for nonzero vectors v, w, and x (be
extra careful!).