# Questions tagged [vector-spaces]

For questions about vector spaces and their properties. More general questions about linear algebra belong under the [linear-algebra] tag. A vector space is a space which consists of elements called "vectors", which can be added and multiplied by scalars

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### Calculate Rotation Matrix to align Vector $A$ to Vector $B$ in $3D$?

I have one triangle in $3D$ space that I am tracking in a simulation. Between time steps I have the previous normal of the triangle and the current normal of the triangle along with both the current ...
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### Is there a quick proof as to why the vector space of $\mathbb{R}$ over $\mathbb{Q}$ is infinite-dimensional?

It would seem that one way of proving this would be to show the existence of non-algebraic numbers. Is there a simpler way to show this?
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### Physical meaning of the null space of a matrix

What is an intuitive meaning of the null space of a matrix? Why is it useful? I'm not looking for textbook definitions. My textbook gives me the definition, but I just don't "get" it. E.g.: I think ...
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### Why are infinitely dimensional vector spaces not isomorphic to their duals?

Assuming the axiom of choice, set $\mathbb F$ to be some field (we can assume it has characteristics $0$). I was told, by more than one person, that if $\kappa$ is an infinite cardinal then the ...
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### Is the vector cross product only defined for 3D?

Wikipedia introduces the vector product for two vectors $\vec a$ and $\vec b$ as $$\vec a \times\vec b=(\| \vec a\| \|\vec b\|\sin\Theta)\vec n$$ It then mentions that $\vec n$ is the vector normal ...
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### Geometric interpretation of $\det(A^T) = \det(A)$

$$\det(A^T) = \det(A)$$ Using the geometric definition of the determinant as the area spanned by the columns, could someone give a geometric interpretation of the property?
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### In categorical terms, why is there no canonical isomorphism from a finite dimensional vector space to its dual?

I've read in several places that one motivation for category theory was to be able to give precise meaning to statements like, "finite dimensional vector spaces are canonically isomorphic to their ...
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### Difference between metric and norm made concrete: The case of Euclid

This is a follow-up question on this one. The answers to my questions made things a lot clearer to me (Thank you for that!), yet there is some point that still bothers me. This time I am making ...
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### In Linear Algebra, what is a vector?

I understand that a vector space is a collection of vectors that can be added and scalar multiplied and satisfies the 8 axioms, however, I do not know what a vector is. I know in physics a vector is ...
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### What kind of matrices are non-diagonalizable?

I'm trying to build an intuitive geometric picture about diagonalization. Let me show what I got so far. Eigenvector of some linear operator signifies a direction in which operator just ''works'' ...
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### What do mathematicians mean by "equipped"?

I am a mathematical illiterate, so I do not know what people mean when they say "equipped". For example, I say that a Hilbert space is a vector space equipped with an inner product. What does that ...
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### Pathologies in module theory

Linear algebra is a very well-behaved part of mathematics. Soon after you have mastered the basics you got a good feeling for what kind of statements should be true -- even if you are not familiar ...
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### Why study finite-dimensional vector spaces in the abstract if they are all isomorphic to $R^n$?

Timothy Gowers asks Why study finite-dimensional vector spaces in the abstract if they are all isomorphic to $R^n$? and lists some reasons. The most powerful of these is probably There are many ...
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### Difference between sum and direct sum

What is the difference between sum of two vectors and direct sum of two vector subspaces? My textbook is confusing about it. Any help would be appreciated.
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### Is every axiom in the definition of a vector space necessary?

Definition: A vector space over a field $K$ consists of a set $V$ and two binary operations $+: V \times V \to V$ and $\cdot: K \times V \to V$ satisfying the following axioms: Commutativity of $+$. ...
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### Reflection across a line?

The linear transformation matrix for a reflection across the line $y = mx$ is: $$\frac{1}{1 + m^2}\begin{pmatrix}1-m^2&2m\\2m&m^2-1\end{pmatrix}$$ My professor gave us the formula above with ...
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### How to understand dot product is the angle's cosine?

How can one see that a dot product gives the angle's cosine between two vectors. (assuming they are normalized) Thinking about how to prove this in the most intuitive way resulted in proving a ...
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### Why are real symmetric matrices diagonalizable?

A matrix is diagonalizable iff it has a basis of eigenvectors. Now, why is this satisfied in case of a real symmetric matrix ?
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