Linear Algebra - Singular Cases

[dontdisturbmycircles]

I am not sure if this belongs in pre-calc or calc and beyond, but I'll put it here.

I need help understanding something, ussually I would not do this, I would sit and think until I understand, but it just isn't happening this time and I need help.

I am self studying some linear algebra just out of curiosity and just started recently. I am using Gilbert Strang's intro to linear algebra 3rd edition.

So basically he is talking about singular cases when a system of linear equations ends up having no solution or infinitely many. Now I understand this would happen if the planes represented by the equations never met (no solution) or met in a line (infinitely many)

He gives the following example.

u + v + w = 2
2u +3w=5
3u+v +4w=6



He says that this system has no solution, which makes sense since if you add the left hands of the first two equations you get the left hand of the last equation but if you add the first two right hands, they don't add up to the last right hand.

My problem is understanding this. He says that if you replaced the 6 in the last equation with a 7, you would get an entire line of solutions (infinitely many). He says the plane "moves to meet the others."... I didn't really understand this so I loaded up maple and tried to understand it by actually plotting the equations. Here is what I got.

http://img84.imageshack.us/img84/2408/nolineincommonpd1.jpg

I don't see a line where they intersect, although I think they most likely do interestect somewhere (but in my oppinion, obviously not everywhere).
I entered it into maple as (smartplot3d[u, v, w])(u+v+w = 2, 2*u+3*w = 5, 3*u+v+4*w = 7).

What does he mean by "the plane moves to meet the others"? :(

ps sorry for the long --- post, couldn't really be shorter though.

 

[Hurkyl]

Well, you can work out exactly where they would intersect if you change that 6 to a 7; it's the line defined by:

u + v + w = 2,
2u + 3w = 5.

(if it helps, you can solve for, for example, v and w as functions of u, and work out a parametric representation for the line)

What does he mean by "the plane moves to meet the others"? :(

Changing the equation changes the plane it defines.

Maybe it will help, maybe not, but you can visualize the constant as a "slider" that, as it varies, translates the plane back and forth. More algebraically, for each a, the equation

3u + v + 4w = a

defines a plane. So, this gives a function:

{real numbers} -> {planes in R³}

 

[dontdisturbmycircles]

Why do you say it is the line represented by:
u + v + w = 2,
2u + 3w = 5.
though?

Is it because the third equation is not really a new equation, but is just derived by adding the first to the second?

By the way I understand that the constant moves the plane around now, it just moves it but the plane stays parallel to its original position.

 

[Hurkyl]

(u, v, w) = (a, b, c) is a solution to

u + v + w = 2
2u +3w=5
3u+v +4w=7

if and only if it is a solution to

u + v + w = 2
2u + 3w = 5.

Since I know the latter system defines a line, I know the former system defines that same line.

 

[dontdisturbmycircles]

Okay well then my maple graph must be incorrect then right? (Quite a possibility as I am not very good with it) The third plane should intersect the first two at the exact same line they intersect at.

 

[Hurkyl]

Do you know how to change the perspective? You only really have a decent perspective on one of the planes.

 

[dontdisturbmycircles]

Here are 3 diff views.

http://img207.imageshack.us/img207/5165/hmm3dr0.jpg
http://img205.imageshack.us/img205/7625/hmm2hx1.jpg
http://img205.imageshack.us/img205/124/hmm1oz2.jpg

Definitely looks like there isn't infinitely many solutions as they don't all interstect in one line, so I am thinking that I am inputting the equations wrong.

 

[dontdisturbmycircles]

I just want to make sure that I understand what they should do. I am very close to understanding this first chapter in the text :P (I want to understand it b4 bed, badly :P). The first two should intersect and the area of intersection would form a line (makes perfect sense) And then in MOST cases, if you add another plane (n equations n unknowns) you could determine a POINT on that line. But in this case the last plane should simply intersect ALL points on that line (like drawing an X and a line through the middle). Right?