Simple conceptual misunderstanding

[jaydnul]

How can, for example, [itex]M_2(R)[/itex] 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, [itex]M_2(R)[/itex] 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 [itex]ℝ^0, ℝ^1, ℝ^2, ℝ^3[/itex]. 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).

 

[blue_raver22]

The concept of a vector space may seem abstract at first, but it is actually a fundamental and concrete concept in mathematics. A vector space is simply a set of objects, called vectors, that can be added together and multiplied by scalars (usually numbers) in a consistent way. This definition allows for a wide range of objects to be considered as vectors, including matrices and sets of solutions to differential equations.

In the case of M_2(R), the set of 2x2 matrices, it can be thought of as a vector space because it satisfies all the properties of a vector space. Matrices can be added together and multiplied by scalars in a consistent way, and this allows for them to be represented as points in a four-dimensional space. This may seem counterintuitive, as we are used to thinking of vectors as arrows in three-dimensional space, but the concept of a vector space allows for a broader understanding of what can be considered a vector.

Similarly, the set of solutions to a linear differential equation can also be thought of as a vector space. This is because the solutions can be added together and multiplied by scalars in a consistent way, just like vectors. This allows for a more abstract understanding of the solutions, rather than just thinking of them as specific functions.

Overall, the concept of a vector space allows for a more general and abstract understanding of mathematical objects that have similar properties to traditional vectors. It is a powerful tool for understanding and analyzing a wide range of mathematical concepts and is essential for many areas of science and engineering. So while it may seem abstract at first, the idea of vector spaces is actually quite concrete and applicable in many contexts.