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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62 votes
10 answers

A linear operator commuting with all such operators is a scalar multiple of the identity.

The question is from Axler's "Linear Algebra Done Right", which I'm using for self-study. We are given a linear operator $T$ over a finite dimensional vector space $V$. We have to show that $T$ is a ...
abeln's user avatar
  • 555
191 votes
21 answers

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 ...
user1084113's user avatar
  • 2,089
30 votes
3 answers

If a field $F$ is such that $\left|F\right|>n-1$ why is $V$ a vector space over $F$ not equal to the union of $n$ proper subspaces of $V$

If $U_1$, $U_2,\ldots,U_n$ are proper subspaces of a vector space $V$ over a field $F$, and $|F|\gt n-1$, why is $V$ not equal to the union of the subspaces $U_1$, $U_2,\ldots,U_n$?
Freeman's user avatar
  • 5,359
175 votes
7 answers

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?
Elchanan Solomon's user avatar
113 votes
3 answers

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 ...
Asaf Karagila's user avatar
  • 393k
73 votes
1 answer

What is the logic/rationale behind the vector cross product?

I don't think I ever understood the rationale behind this. I get that the dot product $\mathbf{a} \cdot \mathbf{b} =\lVert \mathbf{a}\rVert \cdot\lVert \mathbf{b}\rVert \cos\theta$ is derived from ...
Danxe's user avatar
  • 1,685
31 votes
2 answers

Cardinality of a Hamel basis of $\ell_1(\mathbb{R})$

What is the cardinality of a Hamel basis of $\ell_1(\mathbb R)$? Is it deducible in ZFC that it is seemingly continuum? Does it follow from this that each Banach space of density $\leqslant 2^{\...
CesareBorgia's user avatar
45 votes
3 answers

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 ...
coconutbandit's user avatar
61 votes
7 answers

How to find a basis for the intersection of two vector spaces in $\mathbb{R}^n$?

What is the general way of finding the basis for intersection of two vector spaces in $\mathbb{R}^n$? Suppose I'm given the bases of two vector spaces U and W: $$ \mathrm{Base}(U)= \left\{ \left(1,1,...
Cu7l4ss's user avatar
  • 963
109 votes
7 answers

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?
dfg's user avatar
  • 3,851
101 votes
1 answer

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 ...
Ben Blum-Smith's user avatar
57 votes
3 answers

Is there a vector space that cannot be an inner product space?

Quick question: Can I define some inner product on any arbitrary vector space such that it becomes an inner product space? If yes, how can I prove this? If no, what would be a counter example? Thanks ...
Huy's user avatar
  • 6,644
54 votes
3 answers

Every proper subspace of a normed vector space has empty interior

There is a conjecture: "The only subspace of a normed vector space $V$ that has a non-empty interior, is $V$ itself." (here, the topology is the obvious set of all open sets generated by the metric $||...
Somabha Mukherjee's user avatar
43 votes
4 answers

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 ?
aaaaaa's user avatar
  • 2,656
37 votes
2 answers

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 ...
Manos's user avatar
  • 25.8k
112 votes
7 answers

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 ...
VF1's user avatar
  • 1,983
27 votes
1 answer

Prove $\mathbb{Z}$ is not a vector space over a field

This is an exercise from Chapter 3 of Golan's linear algebra book. Problem: Show $\mathbb{Z}$ is not a vector space over a field. Solution attempt: Suppose there is a such a field and proceed by ...
Potato's user avatar
  • 40k
21 votes
5 answers

Is the closure of a convex set again a convex set?

Today I was going through my text book exercise and got hold of the following question. Let $X$ be a normed linear space with norm $\lVert\cdot\rVert$ and $A$ a nonempty convex subset of $X$. Then ...
nimeshkiranverma's user avatar
0 votes
0 answers

How to get the Normal, Tangent, Binormal... to a Polytope?

Context: I am working with polytopes, I am looking for a general way of computing the normal, tangent, bi-normal, tri-normal... etc. to any $n$-polytope, I am well aware a cross-product of linearly ...
linker's user avatar
  • 289
70 votes
15 answers

Dot Product Intuition

I'm searching to develop the intuition (rather than memorization) in relating the two forms of a dot product (by an angle theta between the vectors and by the components of the vector ). For ...
nerdy's user avatar
  • 3,248
53 votes
2 answers

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'' ...
ante.ceperic's user avatar
  • 5,221
48 votes
7 answers

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 ...
Leo's user avatar
  • 1,539
24 votes
3 answers

Geometric interpretation of the multiplication of complex numbers?

I've always been taught that one way to look at complex numbers is as a Cartesian space, where the real part is the $x$ component and the imaginary part is the $y$ component. In this sense, these ...
Justin L.'s user avatar
  • 14.4k
17 votes
6 answers

Sub-dimensional linear subspaces of $\mathbb{R}^{n}$ have measure zero.

I would appreciate it if someone could refer me to a proof (or simply give one here) for the statement in the title. That is: If $k<n$, then every $k-$dimensional subspace of $\mathbb{R}^{n}$ has $...
Serpahimz's user avatar
  • 3,753
4 votes
5 answers

Uniqueness of additive identity element of vector space (or group or monoid)

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. Proof of uniqueness of identity element of addition of vector space ...
DracoMalfoy's user avatar
  • 1,311
80 votes
3 answers

Connections between metrics, norms and scalar products (for understanding e.g. Banach and Hilbert spaces)

I am trying to understand the differences between $$ \begin{array}{|l|l|l|} \textbf{vector space} & \textbf{general} & \textbf{+ completeness}\\\hline \text{metric}& \text{metric space} &...
vonjd's user avatar
  • 8,780
34 votes
1 answer

Vector Spaces and AC

I know that the proof that every vector space has a basis uses the Axiom of Choice, or Zorn's Lemma. If we consider an axiom system without the Axiom of Choice, are there vector spaces that provably ...
user avatar
46 votes
3 answers

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:...
Sadeq Dousti's user avatar
  • 3,261
25 votes
5 answers

Cross product in $\mathbb R^n$

I read that the cross product can't be generalized to $\mathbb R^n$. Then I found that in $n=7$ there is a Cross product: Why is it not ...
Anna's user avatar
  • 1,747
13 votes
1 answer

Vector spaces - Multiplying by zero scalar yields zero vector

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. The following proof is solely based on vector space related axioms. ...
DracoMalfoy's user avatar
  • 1,311
11 votes
3 answers

A vector space over an infinite field is not a finite union of proper subspaces? [duplicate]

Show that if $V$ is a vector space over an infinite field $\mathbb{F}$, then $V$ cannot be written as set-theoretic union of a finite number of proper subspaces.
user219190's user avatar
11 votes
2 answers

Norm induced by convex, open, symmetric, bounded set in $\Bbb R^n$.

Let $A\subset \mathbb{R}^n$ be any bounded, open, convex, and the centre symmetry set having centre at $0$, that is if $x\in A$, then $-x\in A$. Show that $$\|x\| = \inf \{k>0 : x/k \in A \}$$ ...
Derrick's user avatar
  • 119
84 votes
19 answers

How to find perpendicular vector to another vector?

How do I find a vector perpendicular to a vector like this: $$3\mathbf{i}+4\mathbf{j}-2\mathbf{k}?$$ Could anyone explain this to me, please? I have a solution to this when I have $3\mathbf{i}+4\...
niko's user avatar
  • 959
55 votes
2 answers

How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$?

How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$? Here's my attempt: Given a Cauchy sequence $\{q_n\}_{n \in \mathbb{N}}$ in $X/Y$, each ...
Metta World Peace's user avatar
41 votes
2 answers

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 ...
alok's user avatar
  • 3,880
40 votes
2 answers

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 ...
Ester's user avatar
  • 3,047
17 votes
2 answers

Hahn-Banach Theorem for separable spaces without Zorn's Lemma

I was reading about the Hahn-Banach Theorem, its many versions and their proofs. It's known that in the proofs we need Zorn's Lemma. But in the book that I'm reading, the author said if $X$ is a ...
Irddo's user avatar
  • 2,307
8 votes
3 answers

Difference between orthogonal projection and least squares solution

When you find the least squares solution you solve $$A^TA = A\vec b$$ but to find the orthogonal projection into the "subspace" A, you multiply this result (the least squares solution) with the ...
martin's user avatar
  • 437
8 votes
1 answer

A basis for $k(X)$ regarded as a vector space over $k$

Can anyone give an explicit basis of the $k$-vector space $k(X) = \operatorname{Quot}(k[X])$ of rational functions over $k$? The dimension is given by $$\dim_k k(X) = \max(|k|, |\mathbb N|).$$ If $...
Ralph's user avatar
  • 1,820
51 votes
4 answers

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 ...
Gregor's user avatar
  • 993
34 votes
1 answer

Using the Determinant to verify Linear Independence, Span and Basis

Can the determinant (assuming it's non-zero) be used to determine that the vectors given are linearly independent, span the subspace and are a basis of that subspace? (In other words assuming I have a ...
eWizardII's user avatar
  • 773
20 votes
1 answer

Finite-dimensional subspace normed vector space is closed

I know that probably this question has already been answered, but I'd like to present my attempt of solution. Let $(E,\|\cdot\|)$ be a normed vector space and let $F\subseteq E$ be a finite-...
avati91's user avatar
  • 2,857
15 votes
1 answer

Smallest/Minimal bases of a topological space

The smallest possible cardinality of a base is called the weight of the topological space. I was wondering if all minimal bases have the same cardinality, and if every base contains a subset whose ...
Tim's user avatar
  • 47.2k
12 votes
3 answers

How to prove $C_1 \|x\|_\infty \leq \|x\| \leq C_2 \|x\|_\infty$?

I want to prove the following theorem (no idea whether it has a name): Let $V = \mathbb{R}^n$ or $\mathbb{C}^n$ and $\|\cdot\|$ be a norm on $V$. Then, there exist $C_1, C_2 > 0$ such that for all ...
Huy's user avatar
  • 6,644
10 votes
3 answers

Any two bases of a finite dimensional vector space must have the same number of elements.

Prove that any two bases of a finite dimensional vector space must have the same number of elements. By considering the following two bases $$S_1 = \{ \alpha_1, \alpha_2 , \ldots, \alpha_n \},$$ $$...
Taylor Ted's user avatar
  • 3,408
6 votes
1 answer

Is it possible to construct a quasi-vectorial space without an identity element?

I mean if there is any construction that satisfies all the conditions for an vectorial space except it lacks an identity element? This questions was posed to me by a classmate last semester and I have ...
Juan Pablo Santos's user avatar
86 votes
9 answers

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 ...
Paul Lee's user avatar
  • 1,015
62 votes
1 answer

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 ...
dfg's user avatar
  • 3,851
48 votes
6 answers

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 ...
dsd's user avatar
  • 505
34 votes
3 answers

What is the kernel of the tensor product of two maps?

Assume that $f_1\colon V_1\to W_1, f_2\colon V_2\to W_2$ are $k$-linear maps between $k$-vector spaces (over the same field $k$, but the dimension may be infinity). Then the tensor product $f_1\otimes ...
Lao-tzu's user avatar
  • 2,846

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