Supplementary Explanations of a Basis

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It is important to first check out the original basis page for the more rigorous definition of a Basis. If you still don't understand, then come back to this page. This page assumes that one already fully understands the terms "span", "linear independence" and "subspace".

What is a Basis?

From the rigorous definition of a Basis, we know that a group of vectors $ v_1, v_2... v_n $ are defined as a basis of a Subspace V if they fulfill two requirements:

• The vectors span V. In other words, every vector in V can be written as a linear combination of the basis vectors.
• The vectors are linearly independent. In other words none of the basis vectors can be written as a linear combination of the other basis vectors.

Note: Putting it loosely, the "subspace V" is fancy math-speak for a specific collection of vectors.

This previous definition is shamelessly copied from the rigorous definition of a Basis.

However, what does this even mean? Let's start with a conceptual method of understanding this.

Conceptual explanations & analogies

Conceptually, we can analogize the basis to other similar ideas, such as atoms and molecules from chemistry, colors, letters and words from english, and...(MORE STUFF TO BE ADDED)

However, as a starting point, it is possible to think of basis vectors as building blocks and their corresponding vector space V is every possible product.

Chemistry

(Note: todo: insert pictures of chemicals)

Let's analogize everything we know in the abstract magical world of math into the more tangible world of Chemistry.

Arbitrarily, let's call our subspace V as every molecule made of only Carbon and Hydrogen, or in chemical terms every vector in V is a hydrocarbon. And let our "vectors" be molecules and "linearly independent vectors" would just mean that each molecule cannot be made up of other molecules. In the end, all we've done is turn vectors into molecules.

Now, consider this: to make every possible hydrocarbon, you only need two molecules, Hydrogen and Carbon! For example, given an arbitrary hydrocarbon, such as Octane (CH8), you can make Octane with Carbon + 8* Hydrogen! This is common sense, since by definition every hydrocarbon is made of Hydrogen and Carbon. Moreover since every hydrocarbon is just a combination of Hydrogen and Carbon, our "vectors", Hydrogen and Carbon span the "subspace" of Hydrocarbons. Finally, since Hydrogen and Carbon are clearly chemically different, they are also "linearly independent".

So what does this all this analogizing show? That Carbon and Hydrogen are the basis vectors for the subspace of hydrocarbons! In other words, you can imagine basis vectors as this specific type of building block: they are capable of generating every possible vector in the subspace V and are the minimum number of "building blocks" necessary to do so.

This second part is important to note. You only need Carbon and Hydrogen. To make every Hydrocarbon, you could have Carbon, Hydrogen and Methane (CH4); however, basis vectors need to only have the minimum number of vectors, and Methane is unnecessary, so it could and should be removed.

"But wait!" you say. "there are other ways to make every available hydrocarbon! What if you used CH (which actually doesn't exist) and H?"

You're absolutely right. To better express your point, an arbitrary hydrocarbon, like Octane (CH8), can be rewritten as CH +7* hydrogen. And thus comes another point to take note of: basis vectors are NOT unique. There are many different groups of basis vectors that can combine to build every possible vector in our space V. I'll go into this with another analogy.

Hopefully from this, you can understand that Basis Vectors

• can produce every possibility of the given "goal/group" of vectors
• include only the minimum number of vectors
• are NOT unique. There are many sets of basis vectors for each space.

Colors

(note: todo: insert color images)

If you've taken a course in classical physics, you'd know that there are three primary colors: red, green and blue.

If you can't guess already, these three will be our "vectors". I think by now you should understand that vectors can be anything really; you just need to have the imagination for it (surprise! math uses imagination!)

By definition, primary colors can make up every possible color out there. For example, an arbitrary color, like purple, is made of red and blue, or white, which is made up of all three. Moreover, this set of "vectors" is "linearly independent", because none of the colors can be made by adding the other two. You can imagine this by trying to make red out of green and blue. It's pretty obvious that that's impossible. Logically, since they still manage to generate every color and don't have any redundant colors among their own set, then they serve as a basis for all colors. So, these primary colors are a basis for all colors. In other words, by varying the amounts of each, you can make every possible color.

So another way to think of basis vectors is just the reduced form of the original set of vectors; you reduce the number of vectors until you get the minimum amount whose linear combinations can still make the original set.

With this particular analogy, I will show the notion touched upon earlier: that sets of basis vectors are NOT unique.

For those of you in art, you will know that there is another set of primary colors often used to make all colors: yellow, magenta and cyan. Magenta is a pinkish color and Cyan is a light blue color.These three colors also can make every possible color, (arbitrarily, if we take green, we can make this by subtracting yellow from cyan) and they are also "linearly independent" (it's impossible to make Cyan out of Magenta and yellow). So therefore, these three are ANOTHER set of basis vectors.

As pointed out earlier, there's more than one set of basis vectors that can serve as a basis for a set of vectors. Or in these color terms, there is more than one triad of colors that can paint the entire spectrum of colors without having any redundancies.

Throughout history, there are other triads of colors, such as {Red, yellow, and blue} and {Orange, blue, and green}. These also manage to create every color out there.

Moreover, basis vectors have other attributes. First is that the set of them is within the original set. What do I mean? Remember Red, blue and green? They're also colors, and are therefore part of the original set. Second, a set of basis vectors can form every other set of basis vectors. For example, the triad of Cyan, Magenta and yellow can be made by mixing the triad of Red, blue and green (Cyan = blue +green, Magenta = Red + blue, yellow = Green + red with varying amounts of each).

So from this analogy I hope you more clearly understand that basis vectors are:

• not unique
• can form other basis vectors
• are part of the original set that they produce

Physical explanations & examples

Let's say you are given two vectors, $ \begin{pmatrix}1 \\0 \end{pmatrix} $ and $ \begin{pmatrix}0 \\1 \end{pmatrix} $.

We know these two vectors are the columns of the Identity matrix.