# 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 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 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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### 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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### 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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### Effect of elementary row operations on determinant?

1) Switching two rows or columns causes the determinant to switch sign 2) Adding a multiple of one row to another causes the determinant to remain the same 3) Multiplying a row as a constant results ...
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### What are some examples of infinite dimensional vector spaces?

I would like to have some examples of infinite dimensional vector spaces that help me to break my habit of thinking of $\mathbb{R}^n$ when thinking about vector spaces.
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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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### 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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### 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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### 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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### 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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### Inner Product Spaces over Finite Fields

Inner product spaces are defined over a field $\mathbb{F}$ which is either $\mathbb{R}$ or $\mathbb{C}$. I want to know what happens if we try to define them over some finite field. Here's an example:...
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### Relation between rank and number of distinct eigenvalues of a matrix

Let $T : V\to V$ be a linear transformation such that $\dim\operatorname{Range}(T)=k\leq n$, where $n=\dim V$. Show that $T$ can have at most $k+1$ distinct eigenvalues. I can realize that the rank ...
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### A rank-one matrix is the product of two vectors

Let $A$ be an $n\times m$ matrix. Prove that $\operatorname{rank} (A) = 1$ if and only if there exist column vectors $v \in \mathbb{R}^n$ and $w \in \mathbb{R}^m$ such that $A=vw^t$. Progress: I'm ...
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### Why are real symmetric matrices diagonalizable? [closed]

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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### Why it is important for isomorphism between vector space and its double dual space to be natural?

I'm reading the book (by A. Kostrikin) on linear algebra and I feel like I'm really missing something about this idea. I understand the formal proofs of: a) isomorphism between vector space $V$ and ...
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### Proof for triangle inequality for vectors

Generally,the length of the sum of two vectors is not equal to the sum of their lengths. To see this consider the vectors $u$ and $v$ as shown below. By considering $u$ and $v$ as two sides of a ...
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### Change of basis matrix to convert standard basis to another basis

Consider the basis $B=\left\{\begin{pmatrix} -1 \\ 1 \\0 \end{pmatrix}\begin{pmatrix} -1 \\ 0 \\1 \end{pmatrix}\begin{pmatrix} 1 \\ 1 \\1 \end{pmatrix} \right\}$ for $\mathbb{R}^3$. A) Find the ...
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### (Dis)Prove $\sum_{i=1}^n\sum_{j=1}^n{(|x_{i}-x_{j}|-|y_{i}-y_{j}|)^2}\geq 4$

Let $n\ge 4$ and two vectors $x$ and $y$ in $\mathbb{R}^n$ that satisfy $\sum_{i=1}^{n}{x_{i}^2}=\sum_{i=1}^{n}{y_i}^2=1$ $\sum_{i=1}^{n}{x_{i} y_i}=0$ $\sum_{i=1}^{n}{x_{i}}=\sum_{i=1}^{n}{y_i}=0$ ...
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### What Is An Inner Product Space?

As I've understood it, what I've learned is that the dot product is just one of many possible inner product spaces. Can someone explain this concept? When is it useful to define it as something other ...
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### Proof that two bases of a vector space have the same cardinality in the infinite-dimensional case.

I am having a difficulty setting up the proof of the fact that two bases of a vector space have the same cardinality for the infinite-dimensional case. In particular, let $V$ be a vector space over a ...
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### Why is the inclusion of the tensor product of the duals into the dual of the tensor product not an isomorphism?

Let $V$ and $W$ be vector spaces (say over the reals). There is a linear injection $V^* \otimes W^* \to (V \otimes W)^*$ which sends $\sum_i f_i \otimes g_i \in V^* \otimes W^*$ to the unique ...
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### What is the geometric meaning of this vector equality? $\vec{BC}\cdot\vec{AD}+\vec{CA}\cdot\vec{BD}+\vec{AB}\cdot\vec{CD}=0$
I was doing some exercises for linear algebra. One of them was to prove that for any four points $A, B, C, D \in \mathbb{R}^3$ the following equality holds: \overrightarrow{BC} \cdot \overrightarrow{...