Contents
Supplementary Explanations of a Basis
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 analogy
(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 also need to be linearly independent; since Methane can be made with Hydrogen and Carbon, it is redundant and needs to be removed. A sign that a set of vectors is a basis is that any vector you add to it from vector space V would be redundant.
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
Colors Analogy
(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.
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. 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.
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. 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. In fact, there are many possible sets of basis vectors that can still correspond to original 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.
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). Third, they are the maximum number of linearly independent vectors. In this case, Red and green are linearly independent and same with blue and green; however red, green and blue is the maximum number of linearly independent vectors in V I can manage before adding redundant ones.
So from this analogy I hope you more clearly understand that basis vectors are:
- not unique
- can form all other basis vectors
- are part of the original set that they produce
- contain the maximum number of linearly independent vectors.
Food
blabla stuff
So from this analogy I hope you finally understand that basis vectors:
- have only a UNIQUE way of making every vector in set V
- the space constructed by the basis vectors must be EXACTLY V
Physical explanations & examples
Now that we're through conceptual examples, let's get some more physical examples i.e. actually math-related ones. We will go through several properties:
- basic examples of what are sets of basis vectors and what aren't
- the determinant of a matrix of basis vectors
- Dimensions
- the Basis Matrix and its properties
Let's say you are given two vectors, $ \begin{pmatrix}1 \\0 \end{pmatrix} $ and $ \begin{pmatrix}0 \\1 \end{pmatrix} $.
First thing to note is that these two vectors are the columns of the Identity matrix. Moreover, it should be relatively easy to see that every two dimensional vector can be written as a sum of
Conclusion
Basis vectors are, quite simply, the minimum number vectors which can construct another space. For examples problems involving this, see Basis_Problems.
