The fuck is a vector space?

get off Veeky Forums and do some problems you pseud

Think about a collection of cats and dogs you have. If you wanted to map them to a vector space you might create some basis vectors:

( Dog , Cat, weight, age)

The vector space in that order. So a young fat cat would be represented by (0,1,5,2). Aka not a dog, is a cat, average weight, and 2 years old.

The usefulness of the vector space would be based on what operators you use. For instance you could use a common operator like inner product, rejections, or whatever else to analyze things.

That is one example of a vector space. In other cases you would use say a 3d geometric space and use it to imagine 3d translations/rotations etc and calculate them using math.

It's just a way to think about things. The simplest to understand are 2d and 3d euclidean spaces. For large vector spaces like say for idea vectors it's best to understand them as bitmaps almost. AKA the cat/dog example above being 0 or 1.

These I can understand fine
It square matrices that form vector spaces?
I can't understand this, how go I generate a space off of a matrix

so for the space of 2x2 matrices, I think of it as a space where each point is a set of 4 numbers, a b c d that are the elements of a 2x2 matrix

Ok
Does order matter?

how do you mean?

Is (1,5,6,3) different than (5,3,6,1)?
Are all the spaces coordinates in a 4 dimentional system?

A vectorspace is a set of objects with a defined addition and scalar multiplication by some associated field that satisfys a bunch of axioms.
A matrix is a just a linear combination specifically from R_n to R_m for a m by n matrix.

* linear transformation

youtube.com/watch?v=kjBOesZCoqc&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab

You're welcome.

Order matters for the components of vectors.

An nxm matrix is isomorphic to R_(n*m) by element wise addition and scalar multiplication, so in that way nxm matrices form a vectorspace. However, a nxm matrix A also has three associated subspaces: the row space: a subspaces of R_m spanned by the rows, the column space: a subspace of R_n spanned by the columns, and the null space: a subspace of R_m that contains all vectors b such that Ab=0.

a vector space is a collection of objects, which can be added and multiplied by a number (such that these operations satisfy some common-sense properties).
the usual "n-tuples of numbers" form a vector space
the collection of all real-valued functions form a vector space, because functions can be added and multiplied by a number
the collection of all matrices of a given size form a vector space, because matrices can be added and multiplied by a number
the collection of all socks in your drawer can be made into a vector space as long as you specify the rules for addition and multiplication by a number.

Thanks, got it now