What do you say about Linear Algebra? You learn Gaussian elimination and vector spaces and you're done. Except if the textbook your teacher chooses insists that you approach LA from a geometric perspective. Then you must interpret systems of linear equations as hyperplanes and such craziness.
To summarize there are two ways of representing a subspace V Ì Rn:
1. V=Span(v1,…,vk) for appropriate vectors v1,…,vk. This is the generalization of our parametric representation of lines and planes in R3.
2. V= N(A) for an appropriate m x n matrix A. This is the representation by a homogeneous system of linear equations, in which we interpret V as the intersection of the m hyperplanes defined by Ai × x = 0.
To summarize there are two ways of representing a subspace V Ì Rn:
1. V=Span(v1,…,vk) for appropriate vectors v1,…,vk. This is the generalization of our parametric representation of lines and planes in R3.
2. V= N(A) for an appropriate m x n matrix A. This is the representation by a homogeneous system of linear equations, in which we interpret V as the intersection of the m hyperplanes defined by Ai × x = 0.
That's it, enough said.
I took Linear Algebra twice. We won't even talk about why that happened. Ahyhoo, the two courses were totally different. One was impossible, one was easy. Either way, I'm done with it!
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