Linear Algebra: need suggestions for learning

[SticksandStones]

I'm currently taking a Linear Algebra course, and am having some issues.

I can do the mechanics of it, solve the problems and all of that but I don't really get the math theory (perhaps not the best wording) behind it.

Things like "The solutions of Ax=0 has a different plane than Ax=b connected by a vector p" seem to fly over my head.

Do you guys have any suggestions for books I could read to get a better understanding? Right now I'm using "Linear Algebra and it's Applications" by David Lay.

 

[HallsofIvy]

Actually, that's a pretty good text. For that particular question, I would recommend looking at it in the xy- plane. What are the solutions to the equation y- 3x= 0. What are the solutions to y- 3x= 2? Graph them!

 

[SticksandStones]

Ok, so I graphed them and I have two parallel lines? I'm not sure what I'm supposed to get out of it. :(

Am I supposed to be showing that they have no point of intersection? If so, what's the point of "Ax=0 for x = p+tv (where t is any real)" ?

 

[HallsofIvy]

Yes, they are two parallel lines, one going through the origin. They are the two "planes" you mention. Any "vector" draw from one line to the other is the "vector p" that connects them. In particular, if we represent a vector by an arrow drawn from the origin to a point on the line, then the vector from the origin to a point on the line through the origin, added to the vector p, shifted over of course, give a point on the second line.

 

[SticksandStones]

Yes, they are two parallel lines, one going through the origin. They are the two "planes" you mention. Any "vector" draw from one line to the other is the "vector p" that connects them. In particular, if we represent a vector by an arrow drawn from the origin to a point on the line, then the vector from the origin to a point on the line through the origin, added to the vector p, shifted over of course, give a point on the second line.

I get this part, but what confuses me as what the vector p represents in terms of this function. In other words, WHY does anyone care about a vector connecting the two lines? I guess the starting point and the magnitude of the vector would represent the values of "y" for a given "x", but how is this so special as to require this extra work?

EDIT: By the way, thanks for the help. It is appreciated. :)

 

[SticksandStones]

Now I'm really confused. I put the two systems in a matrix and, as I figured they would, it came out to be an inconsistent system.

I really do not see what any of this has to do with anything! :(