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From the rigorous definition of a Basis, we know that a group of vectors <math>v_1, v_2... v_n</math> are defined as a basis of a [[Subspace]] V if they fulfill two requirements: | From the rigorous definition of a Basis, we know that a group of vectors <math>v_1, v_2... v_n</math> are defined as a basis of a [[Subspace]] V if they fulfill two requirements: | ||
| − | *The vectors [[span]] V. | + | *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|linearly independent]]. | + | *The vectors are [[Linearly Independent|linearly independent]]. In other words none of the basis vectors can be written as a linear combination of the other basis vectors. |
This previous definition is shamelessly copied from the rigorous definition of a Basis. | This previous definition is shamelessly copied from the rigorous definition of a Basis. | ||
| − | However, what does this mean? Let's say you are given two vectors, | + | However, what does this even mean? Let's start with a conceptual method of understanding this. |
| − | <math> \begin{pmatrix} | + | |
| − | + | ===Conceptual explanations & analogies=== | |
| − | + | ||
| − | \end{pmatrix} | + | 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) |
| − | </math> | + | |
| + | 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 chemically different 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! 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, | ||
| + | |||
| + | |||
| + | |||
| + | |||
| + | ===Physical explanations & examples=== | ||
| + | Let's say you are given two vectors, | ||
| + | <math> \begin{pmatrix}1 \\0 \end{pmatrix} </math> and <math> \begin{pmatrix}0 \\1 \end{pmatrix} </math>. | ||
| + | |||
| + | We know these two vectors are the columns of the Identity matrix. | ||
Revision as of 14:09, 11 March 2013
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.
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 chemically different 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! 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,
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.
