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MNskating

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- Thread starter MNskating
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In summary, a vector is an object with both magnitude and direction, while a vector space is a more abstract concept that refers to a collection of vectors with certain properties. It is not limited to just 3-dimensional spatial vectors, but can include other mathematical objects. Understanding the concept of a vector space is important in fields such as quantum mechanics, and studying linear algebra is recommended to gain a better understanding.

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MNskating

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andresB

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jtbell

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MNskating said:What is the difference between a vector and a vector space?

I think of it as similar to the difference between a (real) number and the number line; or the difference between a complex number and the complex (Argand) plane.

As andresB noted, the 3-d spatial vectors that we learn about in intro physics classes are only one example of the more abstract mathematical concept of "vector". The set of all 3-d spatial vectors is a vector space, but it's not the only vector space.

Knowing that some set of mathematical objects and their associated operations form a vector space is useful, because you can use your experience with other vector spaces to guide you in working with them. When I was first learning quantum mechanics as an undergraduate, it was a big "aha!" moment for me when I saw that integrals like $$\int{\psi_1^*(x)\psi_2(x)dx}$$ were like vector dot products ##\vec v_1 \cdot \vec v_2##, etc.

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mathman

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bhobba

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MNskating said:What is the difference between a vector and a vector space? I get that a vector is an object with both magnitude and direction, but am confused by the term "vector space". Does a vector space simply refer to a collection of vectors?

You need to study linear algebra. But since this is the quantum physics sub-forum doing it using the bra-ket notation will prepare you for its use in QM:

http://galileo.phys.virginia.edu/classes/751.mf1i.fall02/751LinearAlgebra.pdf

Thanks

Bill

A vector is a mathematical object that represents both magnitude and direction. It is typically denoted as an arrow with a specific length and direction, and can be used to describe quantities such as velocity, force, and displacement.

A vector space is a mathematical structure that consists of a set of vectors and operations, such as addition and scalar multiplication, that can be performed on these vectors. It is a fundamental concept in linear algebra and is used to study geometric and algebraic properties of vectors.

Vectors are individual mathematical objects, while vector spaces are mathematical structures that contain multiple vectors. Vectors can exist without a vector space, but a vector space cannot exist without vectors.

Yes, a vector space can contain only one vector. However, in order for it to be considered a vector space, it must also contain the zero vector (a vector with a magnitude of 0) and follow certain properties, such as closure under addition and scalar multiplication.

Vector spaces are used in many areas of science, including physics, engineering, and computer science. They are particularly useful in representing physical quantities and mathematical models, and are essential in solving problems involving linear systems and transformations.

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