Important changes in the Middle East between the 13th and 15th centuries include the 13 Century end of the First Crusade and the capture/founding of Jerusalem. After that came the rise of the Mongol/Turkish/Ottoman Empire during the 14th Century. The Ottoman transcontinental Empire controlled much of North Africa, Western Asia, and Southeast Europe. This led to the creation of important transcontinental trade routes and a boom of economic trade between the continents.
Pushing the Mongols north and fighting them, and the rebuilding of Beijing caused the Chinese decision to abandon major expeditions.
The answer is most likely A)
The economic crisis in most of the European countries was one of the primary reasons for migration in European countries.
Explanation:
Earlier after the end of second world war, many people migrated from country side to developed countries in search for work, with the intend to get a better life. Many unskilled labors got job in developed countries after this world war.
But when countries started facing economic crisis many workers started migrating to their home countries. There was land shortage and job shortage at this time. Farmers faced crop failure
Another cause of this migration was civil war that took place in Spain.
Most people migrated to seek religious, political and personal freedom.
This migration was also impacted by global conflicts also.
Solution. To check whether the vectors are linearly independent, we must answer the following question: if a linear combination of the vectors is the zero vector, is it necessarily true that all the coefficients are zeros?
Suppose that
x 1 ⃗v 1 + x 2 ⃗v 2 + x 3 ( ⃗v 1 + ⃗v 2 + ⃗v 3 ) = ⃗0
(a linear combination of the vectors is the zero vector). Is it necessarily true that x1 =x2 =x3 =0?
We have
x1⃗v1 + x2⃗v2 + x3(⃗v1 + ⃗v2 + ⃗v3) = x1⃗v1 + x2⃗v2 + x3⃗v1 + x3⃗v2 + x3⃗v3
=(x1 + x3)⃗v1 + (x2 + x3)⃗v2 + x3⃗v3 = ⃗0.
Since ⃗v1, ⃗v2, and ⃗v3 are linearly independent, we must have the coeffi-
cients of the linear combination equal to 0, that is, we must have
x1 + x3 = 0 x2 + x3 = 0 ,
x3 = 0
from which it follows that we must have x1 = x2 = x3 = 0. Hence the
vectors ⃗v1, ⃗v2, and ⃗v1 + ⃗v2 + ⃗v3 are linearly independent.
Answer. The vectors ⃗v1, ⃗v2, and ⃗v1 + ⃗v2 + ⃗v3 are linearly independent.
