Basics of Linear Vector Spaces | Quantum Science Philippines
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Sticky: Basics of Linear Vector Spaces

by CARIEL O. MONTALBAN


In quantum mechanics, I have learned that the wavefunctions, , reside in Hilbert’s space.  What is Hilbert’s space? I guess to answer this question requires exploring the basic  properties of Hilbert’s space.

Hilbert’s space is a linear vector space whose elements, entities or components obey certain rules or axioms.  This means firstly than you can add these elements and the resulting sum is also as a member or entity of  that  space. Secondly, you can multiply the elements with any arbitrary scalar and the product yields something which is also a component of that same space.  Additionally, the operations of addition and multiplication obey definite rules. These rules are called axioms for addition and multiplication.

By means of simple problems discussed below, I illustrate these axioms which are obeyed by a linear vector space and to which the wavefunctions, , of quantum mechanics  belongs.

As a simple example, let us consider the set of all entities of the form where are real numbers. Do these form a linear vector space? First, we have know how these elements are added and how they multiply with scalars. If their addition and multiplication are defined respectively as follows:

;

and

,

we can then verify that the axioms required for a linear vector space are satisfied in this case.

From the addition operation, we can write the null vector of the set as:

.

Also from the multiplication operation, we can then write down the inverse of simply as  .

We can now verify that all four axioms for addition of elements of the set are satisfied.

First Axiom: Commutativity Property

The operation of addition in a linear vector space is commutative; which means that we don’t care about the order in which the elements are added because we always get the same result.  This axiom is written as:

(i)

Our proof is as follows. Let and .

Then,

.

Thus in a linear vector, the addition of vectors is commutative.

Second Axiom: Associative Property

The operation of addition in a linear vector space is associative which means that we don’t care about the order in which two elements are added to the third one because we always get the same result. This axiom is expressed as:

(ii) .

To prove this in the case of the set , we let

Then,

.

Therefore the addition of vectors in a linear vector space is associative.

Third Axiom: Existence of an identity element

The third requirement for a set to be a linear vector space is that the identity element exists. The identity element is defined as

(iii) .

The identity element of the set is therefore none other than the null vector

To show this property, we just apply the definition of addition hence

.

Fourth Axiom: Existence of an inverse

The inverse of a vector should exist in a linear vector space. The inverse is defined by the statement

(iv) .

For the set we can then verify the existence of an inverse as follows:

Examples of non-vector spaces

From the four axioms of addition of linear vector space, we can further make the following observations.

(1) If are required to be positive numbers, we can’t construct a vector space because Axiom (iv) will not be satisfied.

(2) The vectors of the form do not form a linear vector space. To show this, we let

where are all real numbers.

Then by Axiom (i),

.

Thus, does not form a linear vector space. The closure property is clearly violated since

.
About the Author:
CARIEL O. MONTALBAN finished his B.S. in Physics from Mindanao State University-Iligan Institute of Technology (MSU-IIT), Iligan City, Philippines in March 2008 and is now a graduate student of the same university. He hopes to become an active researcher in the field of experimental physics in the future.

24 Responses to “Sticky: Basics of Linear Vector Spaces”

  1. Maryjane Says:

    Hi Cariel. I don’t know if this will help in any way but I guess there is a little typo in the second sentence of your second paragraph. I think the word there is ‘that’ instead of ‘than’. Also, in the third sentence of your fourth paragraph, the verb there is ‘have known’ instead of ‘have know’. Hope this helps in the refinement of your article.

  2. Ancelie Says:

    Hi cariel,

    congrats for the job well done. i like your article. it’s very impressive. we will also make our articles like yours. i am refreshed about linear vector spaces with your article. but is that all? please add more. keep up the good work.

  3. Henrilen Says:

    Hello Cariel

    The article is okey but I think it is nice if you add more discussions on your examples.

  4. Bien Says:

    Cariel!!!

    I agree to the suggestions of Miss Maryjane and I just want to suggest also if you could replace the last equation of your article..instead of “not equal to”, you can have it as “not an element ” of the VECTOR SPACE..

    In MATHEMATICAL NOTATIONS of course..

    Thanks…

  5. Pearl Angel Says:

    I appreciate the way the article is written. It explains the Linear Vector space in the simplest way understandable to all the readers.

  6. simon Says:

    This topic has always been my favorite since I think I understand linear vector space properties well and I thank my excellent mentor for it.

  7. bebelyn Says:

    From physicist point of view, the article is okay and understandable. But as of the whole site, might do something that will really catch the attention of the reader…I mean, if Im not a physicist without knowing some concepts, I might not be interested in reading further…some colors or backgrounds i think might do the magic.

  8. floramie Says:

    Cari hi, the article is very nice but if I were to suggest, maybe it’s much better if the equations were aligned.

  9. Rash Says:

    I guess the article was okay but it would be nice to make it understandable to more audience so that even if the people without so much background in quantum mechanics could appreciate it.

  10. Karen Says:

    Good job! The article is very educational.

  11. John Says:

    This post has been included in the 50th Carnival of Mathematics. Stop by and see the other posts at http://www.johndcook.com/blog/2009/02/27/50th-carnival-of-mathematics/.

  12. Jessica Says:

    hi, I guess the article is fine. Just make a clearer sets of equations because some of it appeared a bit blurry. Thanks! Congrats, by the way, for a job well done!

  13. Rowenchell Says:

    I think it is better to put more examples for the application of the axioms. But the discussions you have cariel is nice. Job well done!!!

  14. Sandra Says:

    Mr. Cariel

    Our knowledge of vector spaces have been refreshed. Thank you.

  15. Make the web Says:

    congrats for the job well done. i like your article. it’s very impressive. we will also make our articles like yours.

  16. Supreme Says:

    Wow! I really wish I could understand this. I have so much respect for the people that do!

  17. snapback Says:

    Wow! I wish I could understand this.

  18. snapback Says:

    I think it is better to put more examples

  19. last kings snapback Says:

    his post has been included in the 50th Carnival of Mathematics

  20. Saon islam Says:

    A simple example of a field is the field of rational numbers, consisting of numbers which can be written as fractions a/b, where a and b are integers, and b ≠ 0. The additive inverse of such a fraction is simply −a/b, and the multiplicative inverse (provided that a ≠ 0) is b/a. To see the latter, note that

    The abstractly required field axioms reduce to standard properties of rational numbers, such as the law of distributivity

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  21. juniperdesert.com Says:

    Passing and adding through worthy subject, i am sure that you focused all aspects with real experiences. It’s nice that you worked well and gave good material over here.

  22. microsoftproblems.com Says:

    i like this page giving me a good briefing on vectors and vector maths. the vector studies are common in quantum mechanics and its application. the mathematical expressions and variables are dealt with in a very good manner that the basics can be easily understood. microsoftproblems.com

  23. mongolian translator Says:

    Hilbert’s space is a direct vector space whose components, elements or parts comply with specific guidelines or adages. This methods firstly than you can include these components and the ensuing entirety is additionally as a part or element of that space. Furthermore, you can reproduce the components with any discretionary scalar and the item yields something which is additionally a part of that same space. Additionally, the operations of expansion and increase obey unequivocal guidelines. These guidelines are called adages for expansion and duplication.

  24. 50th Carnival of Mathematics | John D. Cook Says:

    […] Montalban presents Basics of Linear Vector Spaces posted at Quantum Science Philippines, a blog for science students in the Philippines. Vector […]

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