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.
Answer:
Explanation:
writing drawing arts and crafts stuff like that
Answer:
in a wide area where the is nothing but just grass
Ang manoryalismo, senyoralismo, o senyoryo ay isang makaprinsipyong organisasyon o komunidad na sumibol noong unang panahon lalong lalo na sa gitnang-kanlurang Europa. Ito ay isang pamamaraan ng paghawak ng isang panginoong may lupa ng mga malalawak na lupain. Ang sistemang ito ay hindi rin nagtagal ng mahabang panahon. Isa itong sistemang pang-ekonomiya kung saan ang mga taga-bukid ay nagbibigay ng serbisyo sa isang piyudal na hari, pinuno, o may-ari bilang kapalit ng proteksiyon.
The following are the answers:
12. I once asked Red Cloud if he could recall having ever been afraid, and in reply he told me this story. He was about sixteen years old and had already been once or twice upon the warpath, when one fall his people were hunting in the Big Horn country, where they might expect trouble at any moment with the hostile Crows or Shoshones.
13. Red Cloud had followed a single buffalo bull into the Bad Lands and was out of sight and hearing of his companions. When he had brought down his game, he noted carefully every feature of his surroundings so that he might at once detect anything unusual, and tied his horse with a long lariat to the horn of the dead bison, while skinning and cutting up the meat so as to pack it to camp.
14. Every few minutes he paused in his work to scrutinize the landscape, for he had a feeling that danger was not far off.
