A vector space is a collection of vectors in which you can scale vectors and add vectors together such that the scaling and addition operations satisfy some nice relationships. The 2D and 3D vectors that we are used to are common examples. A less common example is polynomials. It’s hard to think of a polynomial as having a direction and a magnitude, but it’s easy to think of polynomials as elements of the vector space of polynomials.
When talking about vector space, you usually need the “scalar (field)”, and scalars need inverse to be well-defined.
So for integers, the scalar should be integer itself.
Sadly, inverse of integers stops being an integer, from where all sorts of number theoretic nightmare occurs
Instead, integers form a ring, and is a module over scalar of integers.
Start with a list of numbers, like [1 2 3]. That’s it, a list of numbers. If you treat those numbers like they represent something though, and apply some rules to them, you can do math.
One way to consider them is as coordinates. If we had a 3-D coordinate grid, then [1 2 3] could be the point at x = 1, y = 2, and z = 3. You could also consider the list of numbers to be a line with an arrow at one end, starting from the point at [0 0 0] and stopping at the other point. This is a geometric vector: a thing with a direction and a magnitude. Still just a list of numbers though.
Now, what if you wanted to take that list and add another one, say [4 5 6], how might you do it? You could concatenate the lists, like [1 2 3 4 5 6] and that has meaning and utility in some cases. But most of the time, you’d like “adding vectors” to give you a result that maps to something geometric such as putting the lines with arrows end-to-end and seeing what new vector that is. You can do that by adding each element of the 2 vectors. And, almost magically, the point at [5 7 9] is where you’d end up if you first went to [1 2 3] and then traveled [4 5 6] further. We made no drawings, but the math modeled the situation well enough to give us an answer anyway.
Going further, maybe you want to multiply vectors, raise them to exponents, and more? There are several ways to do these, and each has different meanings when you think about them with shapes and geometry.
But vectors are just lists of numbers, they don’t have to be geometric things. [1 2 3] could also represent the coefficients of a function, say 0 = 1x^2 + 2x + 3(x^0). You can still do the same math to the vector, but now it means something else. It models a function, and combining it with other vectors let’s you combine and transform functions just like if they were lines and shapes.
When you get into vectors beyond 3 elements, there’s no longer a clean geometric metaphor to help you visualize. A vector with 100 elements can be used just as well as one with 2, but we can’t visualize a space with 100-dimensions. These are “vector spaces” and a vector is a single point (or rather, points to a point) within them.
Matrices are similar but allow for deeper models of more complex objects.
Very well explained, thank you. I keep forgetting, and am occasionally reminded, that just below the basic math I’m familiar with is a whole other level of advanced math, and just below that is the screaming void.
I don’t understand
A vector space is a collection of vectors in which you can scale vectors and add vectors together such that the scaling and addition operations satisfy some nice relationships. The 2D and 3D vectors that we are used to are common examples. A less common example is polynomials. It’s hard to think of a polynomial as having a direction and a magnitude, but it’s easy to think of polynomials as elements of the vector space of polynomials.
Thanks for clearing that up ❤️
A vector space is when you can:
And get another Thing that’s the same Kind of Thing.
By Thing I mean Vector and by Kind of Thing I mean element of the same Vector Space.
Examples of vector spaces:
Examples of Not Vector Spaces:
Yeah a few of these come with asterisks I’m happy to answer questions but don’t want to argue with pedants.
wow didn’t expect this to be so general. How do integers not fit into the definition ? you can add them together and obtain another integer
When talking about vector space, you usually need the “scalar (field)”, and scalars need inverse to be well-defined.
So for integers, the scalar should be integer itself. Sadly, inverse of integers stops being an integer,
from where all sorts of number theoretic nightmare occursInstead, integers form a ring, and is a module over scalar of integers.deleted by creator
Start with a list of numbers, like [1 2 3]. That’s it, a list of numbers. If you treat those numbers like they represent something though, and apply some rules to them, you can do math.
One way to consider them is as coordinates. If we had a 3-D coordinate grid, then [1 2 3] could be the point at x = 1, y = 2, and z = 3. You could also consider the list of numbers to be a line with an arrow at one end, starting from the point at [0 0 0] and stopping at the other point. This is a geometric vector: a thing with a direction and a magnitude. Still just a list of numbers though.
Now, what if you wanted to take that list and add another one, say [4 5 6], how might you do it? You could concatenate the lists, like [1 2 3 4 5 6] and that has meaning and utility in some cases. But most of the time, you’d like “adding vectors” to give you a result that maps to something geometric such as putting the lines with arrows end-to-end and seeing what new vector that is. You can do that by adding each element of the 2 vectors. And, almost magically, the point at [5 7 9] is where you’d end up if you first went to [1 2 3] and then traveled [4 5 6] further. We made no drawings, but the math modeled the situation well enough to give us an answer anyway.
Going further, maybe you want to multiply vectors, raise them to exponents, and more? There are several ways to do these, and each has different meanings when you think about them with shapes and geometry.
But vectors are just lists of numbers, they don’t have to be geometric things. [1 2 3] could also represent the coefficients of a function, say 0 = 1x^2 + 2x + 3(x^0). You can still do the same math to the vector, but now it means something else. It models a function, and combining it with other vectors let’s you combine and transform functions just like if they were lines and shapes.
When you get into vectors beyond 3 elements, there’s no longer a clean geometric metaphor to help you visualize. A vector with 100 elements can be used just as well as one with 2, but we can’t visualize a space with 100-dimensions. These are “vector spaces” and a vector is a single point (or rather, points to a point) within them.
Matrices are similar but allow for deeper models of more complex objects.
Very well explained, thank you. I keep forgetting, and am occasionally reminded, that just below the basic math I’m familiar with is a whole other level of advanced math, and just below that is the screaming void.