Overview of Linear Algebra

Question

Very new student tackling this course, and I've never been this terrified from Math before. I cannot grasp the meaning of things in Linear Algebra, most of what's stated is either obscure, meaningless, and abstract when I am first tackling them. What sort of advice can you give? I'll be more specific and state my misconceptions.

First and foremost, the simplest question of all; What in the world is a subspace? I am NOT looking for the criteria that makes a subspace so. I want to cling into any fundamental definition. Why for instance should it include the 0 vector? What does a subspace even look like (assuming we are working with R2 and R3)

When I am told that it possess closure under addition and multiplication, I always imagine an infinite plane or a line -- basically; the coordinate space as whole (If that really is the case, then what is the point of defining a subspace).

These lingering thoughts are really dragging me behind...I have a LOT more questions to come, but let this be an impression.

Thanks for Clarifying~

Answer 1

First of all, while it is good to have an intuition and visualization for what mathematical objects are, you cannot remove yourself from the formal definitions. When things get too complicated to visualize, knowing and being able to apply definitions will save you.

That being said, recall that a vector space $X$ is a set which satisfies a long list of axioms. A subspace of $X$ is a subset of $X$ which also satisfies all those axioms. One of the axioms is that a vector space must include a zero vector. Thus, any subspace must also include the zero vector.

A subspace of $\mathbb R^2$ is either just the origin or a line through the origin (or even all of $\mathbb R^2$ is a subspace of itself). A subspace of $\mathbb R^3$ is either the origin, a line containing the origin, or a plane containing the origin (or again, all of $\mathbb R^3$). The reason for defining a subspace is that not all vector spaces can be visualized at $\mathbb R^2$ or $\mathbb R^3$. Some are vastly more complicated. By having a formal definition, we can find properties true of all subspaces, not just planes and lines.

Answer 2

It is interesting that one of the earliest ways of thinking about linearity was also one of the most abstract. Fourier formulated the principle of superposition in a way that very closely follows the definition of linear (vector) space that we use today. Fourier used this principle to formulate his theory of Heat Conduction.

Suppose that you are studying Heat Conduction in some system. You can imagine applying some source of heat $s$, and seeing what happens. This could be a point source of heat, or a general introduction of heat into the system, etc.. You can double the source of heat by applying $2s$. You can add two different sources of heat $s_1+s_2$ and you can form combinations $3s_1+1.2s_2$. Fourier formulated heat conduction using his principle of superposition in a critical way. If $Es_1$ represents the effect on temperature over time due to the application of heat source $s_1$, then his principle was that, if you doubled the source, you would double the effect: $E(2s_1)=2E(s_1)$. And if you added sources, that would lead to adding the effects $E(s_1+s_2)=Es_1 + Es_2$. This was a very abstract principle, but virtual every Physical system has some linear regime where such basic rules apply: double the cause, and you double the effect; add causes and you add their individual effects. The causes will have to be "small" for linearity to apply, but virtually every Physical system has a linear regime. The derivative of Calculus is so important because it gives you a linear regime for the behavior of a function near a point; derivative is a linear approximation, and this generalizes to any number of dimensions.

So, one of the earliest linear space ideas would be the space of causes $s_1,s_2,s_3,\cdots$, and another space would be the space of effects. You can add causes. You can scale causes, etc.. And there is some linear operator $E$ that takes causes to effects through a principle of superposition: double the cause, you double the effect; add two causes and you add their effects.

To think in these terms, you just about have to use the most abstract, modern notion of Vector Space. There was a natural evolution leading to these ideas. And, once these ideas had sufficiently evolved, Vector Spaces because a natural setting for the theory of Quantum Mechanics. Oddly enough, Quantum Mechanics is one of the most linear of the physical models out there, and it requires complex scalars. You have little arrows in the complex plane that spin in time when you look at light, and to understand how to superimpose light, you have to use Complex scalars. Feynman formulated his Quantum Electrodynamics using this understanding.

A subspace is where you consider all the possible causes you can build from a fixed collection of sources using superposition. The corresponding effects would then be the subspace of all possible effects formed by superposition of the corresponding effects.

Answer 3

First I think it works best when you ask specific questions here. But to answer your question, I think the best way to understand linear algebra is to stay abstract. At least that should be your first step. I want to clarify this is only my personal opinion though.

I think lots of the confusion coming from trying to picture different mathematical objects in your head. The picture helps, of course. However, it is still very important to stick with the definitions and working on various counterexamples on your own. Now, to answer some of your questions.

What in the world is a subspace?

Just read the definition, you know it. However, what is a vector? That's a more interesting question.

Why do we want to include zero vector in a subspace?

We don't want to include $0$. Zero vector is always there. As long as you define a subspace as a subset satisfying close under scalar multiplication and addition, $0$ is there. You really cannot get rid of it.

What does a subspace even look like?

In $\mathbb{R}^2$ it looks like line passing through the origin. In $\mathbb{R}^3$ it looks like a plane passing through the origin. In higher dimension, we don't know. If someone knows, please let me know. An interesting question is that why they all have to pass through the origin?

As Bungo mentioned, a line through the origin is also a subspace of $\mathbb{R}^3$, and the origin (single point) is a subspace of both ℝ2 and ℝ3. Also, the space is a perfectly valid subspace of itself.

Answer 4

My advice is to be patient. It's probably not at all clear yet why a subspace is a useful concept, even if you perfectly understand the definition and specific examples in $\mathbb R^2$ and $\mathbb R^3$.

But very soon, you will start learning about linear maps from one vector space to another. The heart of linear algebra is really the study of linear maps between vector spaces, not of vector spaces themselves. This is where subspaces start naturally appearing.

For example, if $f : V \to W$ is a linear map from vector space $V$ to vector space $W$, then the set of all vectors $v$ in $V$ such that $f(v) = 0$ is a subspace of $V$, called the kernel (or null space) of $f$.

And the set of all vectors $w$ in $W$ such that $w = f(v)$ for some $v$ in $V$ is a subspace of $W$, called the image of $f$.

These two subspaces, the kernel and image, are extremely important first steps toward understanding the structure of a linear map. Looking further ahead, you will encounter other spaces which give even more insight: eigenspaces, orthogonal complements, all kinds of cool stuff.

Hopefully this will motivate you more than the dry statement that "a subspace is a subset of a vector space which happens to be a vector space in its own right", which is indeed all it is.