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

learn more… | top users | synonyms

69
votes
4answers
11k views

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?
55
votes
3answers
5k views

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 ...
46
votes
3answers
5k views

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 ...
43
votes
1answer
982 views

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 ...
42
votes
6answers
3k views

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 Hilbert space is a vector space equipped with a inner product. What does that ...
40
votes
13answers
70k views

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 ...
40
votes
6answers
16k views

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 ...
36
votes
2answers
10k views

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 ...
29
votes
4answers
1k views

Given two basis sets for a finite Hilbert space, does an unbiased vector exist?

Let $\{A_n\}$ and $\{B_n\}$ be two bases for an $N$-dimensional Hilbert space. Does there exist a unit vector $V$ such that: $$(V\cdot A_j)\;(A_j\cdot V) = (V\cdot B_j)\;(B_j\cdot V) = 1/N\;\;\; \ ...
28
votes
14answers
1k views

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 ...
26
votes
3answers
618 views

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: ...
25
votes
4answers
755 views

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 ...
24
votes
1answer
1k views

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 ...
23
votes
2answers
10k views

Determinant of transpose?

$$\det(A^T) = \det(A)$$ Using the geometric definition of the determinant as the area spanned by the columns could someone give a geometric interpretation of the property? Thanks!
23
votes
3answers
3k views

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 ...
22
votes
4answers
644 views

Proving an integer is non-negative by showing there is a vector space with it as its dimension.

The other day I attended a lecture on methods to show whether or not a number is an integer. We were given examples of showing it is the number of ways to count something, and to show there exist ...
21
votes
1answer
1k views

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 ...
21
votes
2answers
2k views

Cardinality of a Hamel basis

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 ...
21
votes
4answers
1k views

How to develop an intuitive feel for spaces

I'm a physicist who's currently delving deeper into what I would call more 'hardcore' maths (e.g. FEM and control theory). Every now and then, I come across various spaces, such as vector spaces, ...
20
votes
3answers
101k views

How to tell if a set of vectors spans a space?

I want to know if the set $\{(1, 1, 1), (3, 2, 1), (1, 1, 0), (1, 0, 0)\}$ spans $\mathbb{R}^3$. I know that if it spans $\mathbb{R}^3$, then for any $x, y, z, \in \mathbb{R}$, there exist $c_1, c_2, ...
20
votes
3answers
2k views

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 ...
19
votes
3answers
14k views

How to find basis for intersection of two vector spaces

What is the general way of finding the basis for intersection of two vector spaces? Suppose I'm given the bases of two vector spaces U and W: $$ \mathrm{Base}(U)= \left\{ \left(1,1,0,-1\right), ...
19
votes
4answers
658 views

Why the whole exterior algebra?

So, I've been reading up on multilinear algebra a bit. In particular, I've been reading up on the construction of of the exterior algebra of a finite dimensional vector space $X$, say over ...
19
votes
1answer
1k views

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 ...
18
votes
2answers
5k views

Understanding isomorphic equivalences of tensor product

I get some big picture of tensor and tensor product by reading their Wikipedia articles, and several questions and answers posted before by others. But I cannot figure out how to show the following ...
18
votes
1answer
950 views

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 ...
18
votes
1answer
329 views

Count the number of bases in a subset

Consider $\mathbb{R}^n$ as a vector space over $\mathbb{R}$. Consider the subset $\mathrm{S}^n = \{(x_1,\ldots,x_n) \in \mathbb{R}^n | x_i = 0 \; \mathrm{or} \; 1\;\forall i = 1,\ldots,n\}$. How many ...
18
votes
1answer
304 views

Combinatorics in finite vector space

Let $q$ be a prime power and $V$ a finite $\mathbb F_q$-vector space with two subspaces $I$ and $J$. Let $k$, $a$ and $b$ be non-negative integers. Determine the number of subspaces $K$ of $V$ ...
17
votes
4answers
8k views

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 ...
17
votes
4answers
2k views

Is a vector space over a finite field always finite?

Definition of a vector space: Let $V$ be a set and $(\mathbb{K}, +, \cdot)$ a field. $V$ is called a vector space over the field $\mathbb{K}$ if: V1: $(V, +)$ is a commutative group V2: $\forall ...
17
votes
1answer
13k views

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 ...
17
votes
1answer
1k views

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 ...
16
votes
5answers
9k views

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 $R^n$ when thinking about vector spaces.
16
votes
2answers
19k views

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 ...
16
votes
2answers
683 views

Geometric interpretation of the cofactor expansion theorem

I find the geometric interpretation of determinants to be really intuitive - they are the "area" created by the column vectors of the matrix. Could someone give me a geometric interpretation of the ...
15
votes
4answers
988 views

Vector Spaces: Redundant Axiom?

Question Why are the axioms for vector space independent? More precisely $1x=x$ seems redundant... (I take the axioms from: Wikipedia) Explanation One has for zero vector: ...
15
votes
9answers
2k views

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 ...
15
votes
7answers
2k views

Why a subspace of a vector space is useful

I'm in a linear algebra class and am having a hard time wrapping my head around what subspaces of a vector space are useful for (among many other things!). My understanding of a vector space is that, ...
15
votes
9answers
2k views

Motivation for linear transformations

Sometimes I hate classes for mathematicians. It is not their precision and formality in building the concepts, but they never give a motivation. So in the course my professor started right with the ...
15
votes
2answers
4k views

Why is cross product only defined in 3 and 7 dimensions? [duplicate]

Why $3$ and $7$? I know from some reading that Hurwitz's Theorem explains this, but can someone help me build some intuition behind this or perhaps provide a simpler explanation? It still seems ...
15
votes
4answers
1k views

What really is ''orthogonality''?

I know that we can define two vectors to be orthogonal only if they are elements of a vector space with an inner product. So, if $\vec x$ and $\vec y$ are elements of $\mathbb{R}^n$ (as a real ...
14
votes
8answers
1k views

Zero vector of a vector space

I know that every vector space needs to contain a zero vector. But all the vector spaces I've seen have the zero vector actually being zero (e.g. $\mathbf{0}=\langle0,0,\ldots,0\rangle$). Can't the ...
14
votes
7answers
1k views

Linear Algebra with functions

Basically my question is - How to check for linear independence between functions ?! Let the group $\mathcal{F}(\mathbb{R},\mathbb{R})$ Be a group of real valued fnctions. i.e ...
14
votes
1answer
39k views

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 ...
14
votes
1answer
373 views

Do you need the Axiom of Choice to assert that every real vector space has a norm?

Math people: This question is 95% answered (the first answer) at Does every $\mathbb{R},\mathbb{C}$ vector space have a norm? and Vector Spaces and AC . The questions, answers, and links found there ...
14
votes
1answer
2k views

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'' ...
14
votes
2answers
393 views

Is every vector space basis for $\mathbb{R}$ over the field $\mathbb{Q}$ a nonmeasurable set?

The existence of subsets of the real line which are not Lebesgue measurable can be argued using the Axiom of Choice. For example, define an equivalence relation on $[0, 1]$ by $a \thicksim b$ if and ...
14
votes
2answers
273 views

Infinite direct product of fields.

Let $F$ be a field, and consider the infinite direct product$$F \times F \times F \times F \times \dots,$$i.e. $\prod_{i=0}^\infty F$, i.e. the direct product of a countable number of copies of $F$. ...
13
votes
6answers
414 views

Other guises for the vector space $\mathbb{R}^n$?

One way the vector space $\mathbb{R}^n$ can come up is as the space of polynomials over $\mathbb{R}$ of degree at most $(n-1)$ . Here we have the isomorphism: $$(a_0,a_1,\ldots,a_{n-1}) ...
13
votes
5answers
2k views

Why is one proof for Cauchy-Schwarz inequality easy, but directly it is hard?

Let's say you are in $\mathbb{R}^n$ and you define the norm as $||x||=\sqrt{x_1^2+x_2^2...+x_n^2}$. This we recognize as the usual norm from the inner product: $||x|| = \sqrt{\langle x, x \rangle}$, ...