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axon
10-23-2003, 12:17 AM
here I am, studying for my linear algebra midterm when I stumble upon the following problem:


Consider the vectors cos(x+y) and sinx in C[-pi, pi]. for what values of y will
the two vectors be linearly dependent?

well the linear part is quite easy so I will do that here. But what gets me is the trig afterwards:


W[cos(x+y), sinx] = | cos(x+y) sinx |
|-sin(x+y) cosx |

= cos(x+y)cosx + sin(x+y)sinx = 0

I wont get into the symantics of linear algebra too much, but basically for the vectrs to be linearly dependent that determinant has to equal zero. Now the problem is for what values of y will this equation hold:


cos(x+y)cosx + sin(x+y)sinx = 0


anyone?

axon
10-23-2003, 01:41 AM
damnit, I hate problems like that...they wont let me go to sleep. So after more than an hour I have found the solution. y has to be an odd multiple of ('pi'/2). I found the solution graphically thought, not algebraically...but the latter is for after i get some sleep.

Maybe one of you guys will know the algebraic way to do this?

Zach L.
10-24-2003, 10:01 PM
Well, you could use the fact that:
cos(x+y) = cos(x)*cos(y) - sin(x)*sin(y)
and the fact that sin x is periodic. The conclusion is readily derived from these.

Silvercord
10-27-2003, 07:24 AM
Just wondering, what math are you guys in (are you in college or high school). We don't do anything cool like that in my calculus class, we're still doing stupid limits and continuity, which are important but my class goes too slow

axon
10-27-2003, 11:40 AM
ZachL I figured it out the next morning. The solution is quite trivial, but I blame lack of sleep for not getting it that night.

SilvercordAs my original posts says - it is a Linear Algebra course. A course that every computer scientist has to take, as it is qute similar and very useful to/for a lot of subjects in CS. This course is taken after differential equations (at least in most of the schools that i know).

Zach L.
10-27-2003, 09:00 PM
Originally posted by axon
ZachL I figured it out the next morning. The solution is quite trivial, but I blame lack of sleep for not getting it that night.


From experience, sleep deprivation is good for writing English papers, but not doing math problems. ;)