Linear Algebra: need suggestions for learning

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Discussion Overview

The discussion revolves around understanding concepts in Linear Algebra, particularly the relationship between the solutions of the equations Ax=0 and Ax=b, and the geometric interpretation of these solutions. Participants are exploring the theoretical underpinnings of these concepts and seeking resources for deeper comprehension.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses difficulty in grasping the theoretical aspects of Linear Algebra despite being able to solve problems, specifically mentioning confusion over the relationship between Ax=0 and Ax=b.
  • Another participant suggests visualizing the equations in the xy-plane and recommends graphing them to understand their solutions better.
  • A participant graphed the equations and noted that they are parallel lines, questioning the significance of this observation and the meaning of the vector p connecting them.
  • Further clarification is provided regarding the geometric interpretation of the vector p as a connection between the two lines, but questions remain about its relevance and significance in the context of the equations.
  • A participant mentions confusion after analyzing the systems in matrix form, leading to an inconsistent system, and expresses frustration about the overall relevance of the concepts discussed.

Areas of Agreement / Disagreement

Participants generally agree on the geometric interpretation of the solutions as parallel lines but express differing levels of understanding regarding the significance of the vector p and the implications of inconsistent systems. The discussion remains unresolved as participants continue to seek clarity.

Contextual Notes

Participants are grappling with the theoretical aspects of Linear Algebra, including the geometric interpretation of solutions and the implications of vector relationships. There is an acknowledgment of confusion regarding the relevance of these concepts, particularly in relation to inconsistent systems.

SticksandStones
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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.
 
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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!
 
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)" ?
 

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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.
 
HallsofIvy said:
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. :)
 
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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! :(
 

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