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I am currently learning about linear algebra. To better understand linear algebra and create a network of mathematical concepts, I would like to know what the Big Picture of Linear Algebra is. Which sentence (or couple of sentences) or image (or images) summarize the essence of linear algebra? Hopefully, it'll be easier for me to build my understanding of linear algebra, if I keep this Big Picture in mind. Thanks!

Excuse me, I realized that I didn't explain what I meant with "Big Picture". Personally, the Big Picture are the key ideas (the essence) of a subject or topic. Now, it's not always easy to get at, especially when a subject is broad. However, you can always boil down the subject to a couple of key concepts something simple, that even a student new to the subject can understand. On this basic understanding, the student can then layers of detail.

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    $\begingroup$ „Big Picture“ isn‘t a mathematical notion. So what exactly are you asking? $\endgroup$ Commented Nov 22 at 19:34
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    $\begingroup$ As a mathematician I often find that understanding the “big picture” viewpoint or intuition is one of the key steps in understanding a subject. And for some reason some books don’t explain the big picture clearly enough. We shouldn’t reject this type of question. We should just share our insights with learners. $\endgroup$ Commented Nov 22 at 19:40
  • $\begingroup$ If intuition could be explained or even taught, it would be called „tuition“. $\endgroup$ Commented Nov 22 at 19:42
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    $\begingroup$ We can certainly summarize the big picture ideas and strategies behind proofs or behind entire subjects and such summaries are extremely useful to learners. $\endgroup$ Commented Nov 22 at 19:46
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    $\begingroup$ Go to Gil Strang’s MIT lectures; watch the first video ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/… $\endgroup$ Commented Nov 22 at 21:27

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In linear algebra we study “linear transformations” which are functions $T:V \to W$ which have the special property that $T(x + y) = T(x) + T(y)$ and $T(cx) = c T(x)$ for all $x, y \in V$ and scalars $c$. The spaces $V$ and $W$ are assumed to have just enough structure that these equations make sense (in other words $V$ and $W$ are what we call vector spaces). Often $V = \mathbb R^n$ and $W= \mathbb R^m$.

We would like to understand these types of functions as well as possible. An important strategy is to find a special list of vectors $v_1, \ldots, v_n \in V$ such that understanding what $T$ does to $v_i$ is easy. For example, maybe $T(v_i) = \lambda_i v_i$ for some scalar $\lambda_i$. If we are lucky, then any vector $v \in V$ can be written as a combination of the special vectors $v_i$, and if so then that makes it easier to evaluate $T(v)$. This viewpoint helps us to understand $T$ and do calculations with $T$ easily.

One reason linear algebra is important is that the fundamental strategy of calculus is to take a nonlinear function (difficult) and approximate it locally by a linear function (easy).

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  • $\begingroup$ Linear algebra is not about approximation of non-linear functions, obviously. So the only thing you said about linear algebra is "easy". Helpful? $\endgroup$ Commented Nov 22 at 20:02
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    $\begingroup$ There is a problem in math education which is that some people are such geniuses that they think seeing the big picture is easy, so they don’t bother to explain it. Meanwhile other people (like myself) struggle for years to find the intuition that nobody bothered to explain. So let’s please just explain the big picture viewpoint when students ask for it. Maybe obvious for you but not obvious for others. $\endgroup$ Commented Nov 22 at 20:12
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    $\begingroup$ There is a problem in math education that people don't bother to learn definitions, theorems, proofs, or (ultimate horror!) to do exercises. They want to be explained intuition (impossible by definition of "intuition"!) to avoid all that tedium. Won't work, ever. $\endgroup$ Commented Nov 22 at 20:18
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    $\begingroup$ You might be using a different definition of “intuition”. Just replace intuition with “big picture summary so that I can see the key ideas more clearly”, that is something that can definitely be explained. $\endgroup$ Commented Nov 22 at 20:23
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    $\begingroup$ Another aspect of intuition (the way I use the term) is finding the right illuminating viewpoint that makes a topic seem more clear. Often just finding the right viewpoint sheds light on a whole subject. $\endgroup$ Commented Nov 22 at 20:31

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