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

In summary, vector spaces are important in math because they allow for a linear approach to problem solving, which can be applied to various scenarios such as geometry, systems of equations, differential equations, polynomials, and more. By studying vector spaces, one can gain knowledge that can be applied to a wide range of problems.
  • #1
Howers
447
5
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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  • #2
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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  • #3
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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  • #4
lolol

I just upgraded you from my favorite mathematician to my favorite human being.
 
  • #5
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.
 

1. What is a Vector Space?

A Vector Space is a mathematical structure that consists of a set of elements (called vectors) and operations of addition and scalar multiplication. These operations follow specific properties, such as closure, commutativity, associativity, and distributivity, making it possible to perform mathematical calculations within the space.

2. What are the properties of a Vector Space?

The properties of a Vector Space include closure, commutativity, associativity, and distributivity. Closure means that the result of any operation within the space is also an element of the space. Commutativity means that the order of operations does not affect the result. Associativity means that the order of operations can be changed without affecting the result. Lastly, distributivity means that scalar multiplication can be distributed over vector addition.

3. What is a Subspace?

A Subspace is a subset of a Vector Space that also follows the properties of a Vector Space. This means that the elements of the subspace can be added and multiplied by scalars without leaving the subspace. Subspaces are useful in simplifying calculations within a larger Vector Space.

4. How do you determine if a subset is a Subspace?

To determine if a subset is a Subspace, it must satisfy three conditions: 1) the subset must contain the zero vector, 2) the subset must be closed under addition, and 3) the subset must be closed under scalar multiplication. If all three conditions are met, then the subset is a Subspace of the original Vector Space.

5. What is the basis of a Vector Space?

The basis of a Vector Space is a set of linearly independent vectors that span the entire space. This means that any vector within the space can be written as a unique linear combination of the basis vectors. The number of basis vectors is known as the dimension of the Vector Space. The basis is a fundamental concept in Vector Spaces as it allows for the representation and manipulation of vectors in a more manageable way.

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