Revision as of 15:31, 11 March 2013

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, 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

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 Octane (CH8), you can make this 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 doesn't exist) and H?"

You're absolutely right. To express your point, Octane (CH8) which was written as Carbon +8*hydrogen 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

I am sure everyone knows what colors are.

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.