Simple conceptual misunderstanding

[jaydnul]

How can, for example, $M_2(R)$ have four dimensions? What I mean is how can a 2x2 matrix be considered a vector? Also how can the set of solutions to a linear differential equation be a set of vectors? Or are these examples supposed to be the idea of vector spaces applied outside the realm of actual vectors? Because vector spaces are introduced in the book as being abstract but up until now, I've thought they were pretty concrete. Is this where the idea of vector spaces becomes abstract?

 

[Mark44]

How can, for example, $M_2(R)$ have four dimensions? What I mean is how can a 2x2 matrix be considered a vector? Also how can the set of solutions to a linear differential equation be a set of vectors? Or are these examples supposed to be the idea of vector spaces applied outside the realm of actual vectors? Because vector spaces are introduced in the book as being abstract but up until now, I've thought they were pretty concrete. Is this where the idea of vector spaces becomes abstract?

A vector is anything that belongs to a vector space. Your text should have a definition for the axioms that must be satisfied for a vector space. One of the basic ideas is that if u and v are vectors in the vector space, then u + v is also in that space. If k is a constant, then ku is in that vector space. We say that the vector space is closed under vector addition and scalar multiplication.

Since there are four entries in a 2X2 matrix, it can be considered to be similar to a vector in R⁴.

A function space is defined in almost the same way as a vector space. If f₁ and f₂ are solutions of a differential equation, then f₁ + f₂ will also be a solution of that diff. equation, as will k*f₁.

 

[jaydnul]

Tell me if this is a correct statement:

Almost all vector spaces are abstract other than $ℝ^0, ℝ^1, ℝ^2, ℝ^3$. We use the rules we have learned to be true for these vector spaces, and apply them to more abstract problems, like the set of solutions to a differential equation?

Thanks for the help btw!

 

[TheOldHag]

I'm learning too. But here is my two cents. Vectors are abstract entities over a field (i.e. real numbers) and that satisfy certain axioms likely to be mentioned in your book. There is nothing different between a list of 4 numbers arranged in 1 row of 4 columns as opposed to a list of 4 numbers arranged in 2 rows of 2 columns when considered as vectors within a vector space. They obey the axioms. Furthermore, there is a 1-1 and onto mapping between the two by mapping the linear list into 2 rows, 2 columns. You still need ((1,0),(0,0))((0,1),(0,0))((0,0),(1,0))((0,0),(0,1)) to span the space as you would in a space spanned by vectors of the forms (1,0,0,0)(0,1,0,0)(0,0,1,0)(0,0,0,1). That is why your space is 4 dimensional.

I think you may be hung up on the visual representation of a vector as a list of number when that is just the most natural way to represent it for us to grasp it intuitively and most common applications of vector spaces are of that nature since it is usually real numbers that we are interested rather than exotic entities (unless I suppose you are into mathematics more advanced than I'm familiar with).