Why are vector spaces and sub-spaces so crucial in math?

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Howers
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What exactly is so special about them?

What makes a set of vectors that are closed under addition/scalar multiplication and contain 0 so important in math? I've worked through many examples and always wonder... what do these rules mean.
 

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Because many interesting things have a vector space structure, and many interesting problems can be formulated as a linear algebra problem. So if you study them in general, that knowledge can be applied to all of these different scenarios.

In your course, you will probably see examples involving geometry, systems of equations, differential equations, polynomials, and maybe even other things.

Basically, you are simply continuing your algebra courses from high school -- you're simply progressing beyond the boring case where you're only manipulating real or complex numbers.
 
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is the rule (f+g)' = f' + g' useful in calculus?

do little brown bears go poopoo in the woods?

is W a moron?

am i a tedious old ****?if you answer yes to any of these then vectors spaces are GOOD for you.
 
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I think you should be asking not why is "a set of vectors that are closed under addition/scalar multiplication and contain 0 so important in math" but why vectors themselves are important.

"Linear Algebra" encapsulates the whole concept of "linearity"- that we can break a problem into pieces, solve each piece, and then put them together for a solution to the original problem. You can't do that with "non-linear" problems. That's why vector spaces are important and sub-space are, of course, vector spaces. Arbitrary sets of vectors are not vector spaces.