Sounds like your issue is with the different representations- try watching this:
http://blip.tv/play/gbJX2ZlGjvMg
Specifically, pay attention to the part when he talks about "Row Picture" vs. "Column Picture".
Hi, i'm having trouble coming to an intuitive (spatial) understanding of what happens when a set of linear equations is written as a vector equation. For example, if you have three linear equations with two unknowns, this corresponds (geometrically) with three lines in the x,y plane ..right? However, viewed as a vector equation, we have two vectors in R^3 (multiplied by the respective variables) set equal to another vector in R^3. Now, instead of being in two dimensions, it seems like we've moved into three dimensions. Is this just a different geometric way of looking at the same thing?
Also, think about a system of two linear equations in three variables. We have two planes in three dimensions. However, viewed as a vector equation, we have three vectors in R^2 (Multiplied by the respective variables) set equal to another vector in R^2. How can we say these vectors are in R^2 when we're dealing with planes in three dimensions?
I'm using David Lay's linear algebra and its applications, if you have any page references.
Sounds like your issue is with the different representations- try watching this:
http://blip.tv/play/gbJX2ZlGjvMg
Specifically, pay attention to the part when he talks about "Row Picture" vs. "Column Picture".