Vectors in elementary physics vs vectors in linear algebra

Question

I am taking a linear algebra course as well as an introductory physics course, and I am confused about how vectors in each are the same and different.

First, why is it that in physics, magnitude and direction of vectors are emphasized while in linear algebra components of vectors are emphasized? Also, for elementary physics (kinematics, dynamics) is $\Re^{2}$ and $\Re^{3}$ the vector space from which we operate? Finally, in physics, it is emphasized that vectors are particularly useful because they are not tied to any one coordinate system. What does this mean exactly? In linear algebra we always write vectors emanating from the origin of the coordinate system, so it would seem as though the vectors are tied to the coordinate system.

Answer 1

Physics uses mathematical models to describe physical observations. You will meet this if you continue studying physics.

The mathematics is used rigorously with all its definitions. What makes it a model is a set of "assumptions", "postulates", "principles" which pick up a subset of all the space the mathematical functions can range. I will try to demonstrate with your "vectors", showing that they model observations given some assumptions.

You ask:

First, why is it that in physics, magnitude and direction of vectors are emphasized while in linear algebra components of vectors are emphasized?

Because vector algebra is used in physics to model impacts, for example, which by observation depend on magnitude and direction. In other uses of vectors the components are emphasized , but you have to continue in physics studies to encounter examples.

Also, for elementary physics (kinematics, dynamics) is R2 and R3 the vector space from which we operate and do our calculations?

We use the real numbers for end results of calculations. In special models complex numbers are important because of the simplicity of the form, but the end result of calculations has to be in real numbers.

Finally, in physics, it is emphasized that vectors are particularly useful because they are not tied to any one coordinate system. What does this mean exactly?

It means that the origin of the vectors is assumed according to the experimental set up that needs to be modeled.

In linear algebra we always write vectors emanating from the origin of the coordinate system, so it would seem as though the vectors are tied to the coordinate system.

The vectors used in physics assume as a coordinate system for the vectors the point where the vector is used to describe a force, or a velocity depending in the model. The origin could be a function of the coordinates, i.e a moving vector.

Answer 2

In the linear algebra found in Michael Artin's Algebra, coordinates are not emphasized*.

You have a field (like $\mathbb{R}$ or $\mathbb{C}$), and your vector space (like $\mathbb{R}^n$ or $\mathbb{C}^n$) which you can multiply by elements of your field. This has to satisfy some big list of axioms, but no axiom mentions coordinates. $\mathbb{R}^n$, consisting of lists of real $n$-tuples, is an example of a vector space, but you could imagine defining it without making use of coordinates. You can define the dimension of the vector space without making use of coordinates either: the dimension is the size of the largest possible set of linearly independent vectors. So there's no coordinate dependence here.

In linear algebra, you can continue using abstract definitions. You care about linear operators on vector spaces. Linear maps have their own abstract definition. Once you choose a basis, composition of linear maps turns into matrix multiplication! So there doesn't have to be any coordinate dependence here either.

But even with all of this coordinate-free stuff, we still don't have the "coordinate-free" nature that physicists care about. We actually have to throw geometry in to get anything useful for physics. In classical mechanics, we throw in Euclidean geometry.

The simplest way to throw in Euclidean geometry is to... use coordinates. We could say that vectors $(x,y)$ have length $\sqrt{x^2+y^2}$ (Note: normally this is the pythagorean theorem and is a "theorem". Here it's an axiom), and that the angle between two vectors satisfies $(v_x,v_y)\cdot(w_x,w_y)=v_x w_x+v_y w_y=\|v\|\|w\|\cos(\theta)$. This turns the vector space into an inner product space, but we still don't have quite enough for physics, just yet. To get something more physical: dictate that all of our physical laws have to be invariant under transformations which leave all lengths and angles unchanged.

Finally, under the structure of a vector space, the Euclidean inner product, and the demand that physics should be unchanged under all linear maps that leave the inner product conserved, we have something resembling a vector in physics.

The moral is that it's a pain in the butt to spell out all of the mathematical assumptions. The vectors in mathematics are precisely anything that satisfied the vector axioms I linked above. The vectors in physics have tons of extra structure on them - like, for velocity vectors in classical mechanics, Euclidean geometry. However this doesn't necessarily have anything to do with coordinates.