Assume that all vectors are non-zero vectors in the statements below. Decide whether the following are true in all cases or false in at least one case.

pretty sure they’re all true

Step-by-step explanation:

1. Should be obvious once you realize that if they all are in the same plane, one of them is a linear combination of the other two (think about a mini R^2 situation on the plane)

2. If x,y,z are linearly independent, they all are in the same plane, so z can be written in terms of x and y. (Above reasoning again.)

3. Should be intuitively clear, but rigorously just do the contrapositive. If v1,v2 are linearly dependent, av1 + bv2 = 0 for some a,b, so you can do av1 + bv2 + 0v3 = 0 which means v1,v2,v3 are dependent as well.

4. Intuitively they all are in the same plane. But if you want to be rigorous, you can choose, for example, 5v1 + 0v2 + 0v3 = 0, so they are linearly dependent.

5. What this is really saying is there is no scalar k such that v1 = kv2. It should be clear they are independent, but again if you want to be rigorous, suppose (for contradiction) there are a,b such that av1 + bv2 = 0. Then v1 + (b/a)v2 = 0, so v1 = -(b/a)v2. But this violates the “k” thing we said before! Hence a contradiction, so such a combination is not possible and thus they are linearly independent.

this story was prepared by jim

mccullaugh in los angeles and

al stewart in new york.

new york the colossal success of "top gun" appears to have dramatically altered madison avenue's

perception of corporate sponsorship

of home video projects.

the paramount home video re- lease, featuring a 60- second diet

pepsi spot, was by no means the

first prerecorded video to carry an advertiser's plug, but it was the

first major theatrical release to do so. in the aftermath of the title's

success, paramount is widely ex- pected to ink a similar deal for

"crocodile dundee" or "star trek

iv" or both.

corporate america's new attraccds push british

recording sales

to 19% gain

london fueled by the compact

disk boom, the value of u.k. manu- facturer shipments of recordings

rose 19% in the 12 -month period ended march 1987, according to the

british phonographic industry.

total value of u.k. shipments for

the period was $755 million for all

configurations. (all value figures

are based on an exchange rate of

$1.68 to the pound sterling.)

cd sales totaled 10.7 million units

worth $121.3 million- compared

with 3.7 million units worth $39.5

million in the prior 12 months -and

accounted for two -thirds of the

overall gain in trade value. at present growth rates, says the bpi, cd

sales for calendar 1987 will reach 20

(continued on page 79)

step-by-step explanation:

1) is right

step-by-step explanation: