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 ...

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6
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2answers
2k views

Finite fields as vector spaces

I'm having great difficulty understanding this topic. Can someone concretely explain what it is meant by thinking of $GF(q^2)$ ($q$ a prime power) as a two-dimensional vector space over its subfield ...
5
votes
1answer
1k views

What is the difference between Cartesian and Tensor product of two vector spaces

In particular, how is it that dimension of Cartesian product is a sum of dimensions of underlying vector spaces, while Tensor product, often defined as a quotient of Cartesian product, has dimension ...
5
votes
0answers
138 views

Why is the radical of a Clifford algebra generated by the kernel of the associated symmetric form? [duplicate]

I was recently reading through Jacobson's Basic Algebra. I got to the section on Clifford algebras, and have the following question. Let $Cl_\omega$ be the Clifford algebra with bilinear symmetric ...
3
votes
2answers
74 views

Alternative definition for span and proving it is equivalent to the most common one

This is a question related to something that I asked here about this alternative definition of span. User hardmath has helped me a lot! Therefore, I can't still understand how to prove the equivalence ...
3
votes
2answers
59 views

Function for diagonalizing a vector.

I was playing around whith the idea of what operation (function) should I perform (apply) over a vector $\mathbf{a} = (a_1,a_2, \ldots, a_N)^T \in \mathbb{R}^N$ to come up with the following matrix: ...
3
votes
5answers
239 views

Dimensions: $\bigcap^{k}_{i=1}V_i \neq \{0\}$

Let $V$ be a vector space of dimension $n$ and let $V_1,V_2,\ldots,V_k \subset V$ be subspaces of $V$. Assume that \begin{eqnarray} \sum^{k}_{i=1} \dim(V_i) > n(k-1). \end{eqnarray} To show that ...
3
votes
2answers
595 views

How to prove that two non-zero linear functionals defined on the same vector space and having the same null-space are proportional?

Let $f$ and $g$ be two non-zero linear functionals defined on a vector space $X$ such that the null-space of $f$ is equal to that of $g$. How to prove that $f$ and $g$ are proportional (i.e. one is a ...
1
vote
3answers
101 views

Show that $T\to T^*$ is an isomorphism (Where T is a linear transform)

I think I solved it, but I used a dirty trick, I'd like someone to review it, that would be great. Let $X,Y$ be linear spaces over field $F$. and $T:X \to Y$ a linear transformation.For each $T$ we ...
1
vote
1answer
106 views

Proof of $(u.v)=(u_{x}.v_{x}+u_{y}.v_{y}+u_{z}.v_{z})$, assuming $(u.v)=|u||v|cos\theta$.

I would really love any sort of proof of this. I have a very elementary geometric proof for $R^{2}$. That's mainly because I can easily represent $cos\theta$ in the form of ...
1
vote
3answers
4k views

How to find an intersection of a 2 vector subspace?

Assuming we have 2 subspaces, $\mathbb W$ and $\mathbb U$ of $\mathbb V$. how to get thier intersection?
1
vote
1answer
332 views

Visualization of 2-dimensional function spaces

As a follow-up question to what is the norm measuring in function spaces I just had an idea: How about visualizing function spaces as normal planes. What I have in mind is to have an orthogonal ...
1
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3answers
3k views

What is the norm measuring in function spaces

In spatial euclidean vector spaces norm is an intuitive concept: It measures the distance from the null vector and from other vectors. The generalization to function spaces is quite a mental leap (at ...
14
votes
1answer
360 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 ...
9
votes
5answers
11k views

How to solve this to find the Null Space

What I did : I put this into reduced row echleon form: $$\begin{bmatrix} 1 & -2 & 2 & 4 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 ...
8
votes
1answer
324 views

Cross Product Intuition

I know the cross product between a vector $a$ and a vector $b$ is just a vector whose magnitude is the product of magnitude of $b$ times the magnitude of the perpendicular component of $a$ in relation ...
7
votes
2answers
377 views

Localization does not commute canonically with infinite direct products

Let $S=\mathbb{Z}-\{0\}$. Show the existence or nonexistence of isomorphism between $S^{-1}\prod_{1}^{\infty}\mathbb{Z}_{i}$ and $\prod_{1}^{\infty}\mathbb{Q}_{i}$ as $\mathbb{Q}$-vector spaces. ...
6
votes
2answers
106 views

Does $\mathbb{R}^n$ have a real vector space structure with dimension other than $n$?

Can we define a vector space structure on $\mathbb {R}^n$ other than usual scalar multiplication and usual addition such that the dimension of $\mathbb {R}^n$ over $\mathbb {R}$ is not $n$ but some ...
6
votes
1answer
327 views

Is there a difference between abstract vector spaces and vector spaces?

I am following my Oxford syllabus and my next step is abstract vector spaces, in my linear algebra book I've found vector spaces. I've searched a little and made a superficial comparison between ...
6
votes
3answers
45k views

Find the equation of the plane passing through a point and a vector orthogonal

I have come across this question that I need a tip for. Find the equation (general form) of the plane passing through the point $P(3,1,6)$ that is orthogonal to the vector $v=(1,7,-2)$. I would ...
6
votes
1answer
8k views

Properties of a matrix whose row vectors are dependent

When a column vector in a matrix is a made up of "combination" of its other column vectors, it is said to be linearly dependant. Say... $$ A=\begin{bmatrix} 2 & 1 & 0\\ 4 & 5 & ...
5
votes
2answers
81 views

Confused about Euclidean Norm

I am trying to understand that the Euclidean norm $\|x\|_2 = \left(\sum|x_i|^2\right)^{1/2}$ is in fact a norm and having trouble with the triangle inequality. All the proofs I have referred to ...
5
votes
2answers
152 views

Does $(x,f(x),\cdots,f^p(x))$ is linearly dependent over $E$ implies $(id, f, …, f ^ p)$ is linearly dependent over $\mathcal{L}(E)$?

Here is the original (classic I think) problem I had encored: if $(x,f(x))$ is a linearly dependent family of $E$ (a vector space) for all $x\in E$, then the family $(id,f)$ is linearly dependentt ...
5
votes
3answers
695 views

Why a non-diagonalizable matrix can be approximated by an infinite sequence of diagonalizable matrices?

It is known that any non-diagonalizable matrix, $A$, can be approximated by a set of diagonalizable matrices, e.g. $A \simeq \lim_{k \rightarrow \infty} A_k$. Why this is true? Note: I was faced with ...
5
votes
2answers
502 views

Sparse basis for linear subspace

Suppose I have a linear subspace of some vector space, e.g. described as the column space of some big matrix. How would I algorithmically find a basis of that same subspace where the basis matrix is ...
5
votes
1answer
708 views

If $X$ is infinite dimensional, all open sets in the $\sigma(X,X^{\ast})$ topology are unbounded.

As in the title, if $X$ is infinite dimensional, all open sets in the $\sigma(X,X^{\ast})$ topology are unbounded. The $\sigma(X,X^{\ast})$ topology is the weakest topology that makes linear ...
4
votes
2answers
90 views

If $A$ is a complex matrix of size $n$ of finite order then is $A$ diagonalizable ?

Let $A$ be a complex matrix of size $n$ if for some positive integer $k$ , $A^k=I_n$ , then is $A$ diagonalizable ?
4
votes
1answer
161 views

If the expectation $\langle v,Mv \rangle$ of an operator is $0$ for all $v$ is the operator $0$?

I ran into this a while back and convinced my self that it was true for all finite dimensional vector spaces with complex coefficients. My question is to what extent could I trust this result in the ...
4
votes
2answers
223 views

Tangent space and tangent vectors

As I have heard, tangent vector to a smooth manifold $M$ in $p \in M$ is the operator $D_{\xi}$:$f \to D_{\xi}f$, where $f$ is a smooth function $f: M \to R$, with the following properties: ...
4
votes
2answers
343 views

why symmetric matrices are diagonalizable?

A matrix is diagonalizable iff it has a set of eigen vectors which are linearly independent. Now, why is this satisfied in case of symmetric matrix ?
4
votes
1answer
80 views

Exact sequence arising from symplectic manifold

Let $M$ be a symplectic manifold, why is the following sequence exact? $$0\to \mathbb{R} \to C^\infty (M)\to A\to 0$$ Here $A$ is the set of global Hamiltonian vector fields.
4
votes
1answer
192 views

$V=W_1\oplus\cdots\oplus W_k$ if and only if $\dim(V)=\sum{\dim(W_i)}$

If $W_1,\dots, W_k$ are subspaces of a finite dimensional vector space $V$ such that $W_1+\cdots+W_k=V$, and I want to show that $V=W_1\oplus\cdots\oplus W_k$ if and only if $\dim(V)=\sum{W_i}$, then ...
4
votes
3answers
150 views

$T:V\rightarrow W$ such that $N(T)=V'\subset V$ and $R(T)=W'\subset W$

How should one prove that there exists a linear map $T:V\rightarrow W$ such that $N(T)=V'\subset V$ and $R(T)=W'\subset W$ if $\dim(V')+\dim(W')=\dim(V)$, where $V$ and $W$ are finite-dimensional ...
4
votes
3answers
598 views

How do you construct the quaternion and the multiplication rules, like Hamilton did?

So, I understand complex number multiplication, and how it represents $2D$ rotations. What I don't understand is, how you add two more imaginary numbers $j$ and $k$, and get $3D$ rotations. I believe ...
4
votes
1answer
2k views

Dimension of Vector Space (Polynomial)

I was asked by a friend to: "Find the dimension of the vector space consisting of all polynomials in $n$-variables of degree at most $k$".Now, my response to him was that since the basis consists of ...
3
votes
1answer
532 views

Find normal vector of circle in 3D space given circle size and a single perspective

I don't really know what to search up to answer my question. I tried such things as "ellipse matching" and "3d circle orientation" (and others) but I can't really find much. But anyways... I have ...
3
votes
2answers
362 views

What is (fundamentally) a coordinate system ?

Consider the following construction of vectors and points. Let's start with a vector space, or more specifically a coordinate space $F^N$ over a field $F$ and of $N$ dimensions. The elements of this ...
3
votes
3answers
2k views

Find a basis for the vector space of symmetric matrices with an order of $n \times n$ [duplicate]

Find a basis for the vector space of symmetric matrices with an order of $n \times n$ This is my thought: by definition of symmetry, $a_{i,j}=a_{j,i}$. Therefore, the basis should consist ${n^2-n} ...
3
votes
1answer
590 views

Algebraic complements in vector space of functions without the axiom of choice

The axiom of choice is equivalent to the statement that every subspace $U$ of every vector space $V$ has an algebraic complement, i.e. another subspace $W$ that has a trivial intersection with the ...
3
votes
4answers
6k views

Why does cross product tell us about clockwise or anti-clockwise rotation?

Wikipedia link for Cross Product it talks about using cross product to determine if 3 points are in clockwise or anti-clockwise rotation. I'm not able to visualize this or think of it in terms of ...
3
votes
3answers
899 views

Induced Exact Sequence of Dual Spaces

So given a short exact sequence of vector spaces $$0\longrightarrow U\longrightarrow V \longrightarrow W\longrightarrow 0$$ With linear transformations $S$ and $T$ from left to right in the ...
3
votes
3answers
2k views

Angle between two vectors?

I have been taught that the angle between two vectors is supposed to be their inner product. However, the book I'm reading states: Recall that the angle between two vectors $u = ...
3
votes
3answers
221 views

Possible proof for the relation involving matrix trace

Suppose a diagonal matrix $D\in\mathbb{R}^{n\times n}$ is given, with all its entries $d_{ii}\geq0$, for all $i$. Is it possible to prove ...
3
votes
1answer
976 views

How many different basis' exist for an n-dimensional vector space in mod 2?

Imagine a 2-dimensional vector space in $\mathbb{Z} /2 \mathbb{Z}$. The only possible basis' are $$ \left(\begin{array}{c} 1\\ 0 \end{array}\right), \left(\begin{array}{c} 0\\ 1 ...
3
votes
1answer
227 views

$4$ idempotent operators $E_1,E_2,E_3,E_4$ $:V\to V$ such that $E_1+E_2+E_3+E_4=I$ but don't partition the identity

Let $V$ be a vector space over $F$ such that $charF \neq 2$ Can anyone help me think of $4$ idempotent operators $E_1,E_2,E_3,E_4$ $:V\to V$ such that $E_1+E_2+E_3+E_4=I$ but $\{E_1,E_2,E_3,E_4 \}$ ...
2
votes
2answers
70 views

Automorphism of algebraic closure $\overline{{\bf F}}_p$.

Problem : I want to give an concrete example of automorphism of $\overline{{\bf F}}_p$ which fixes ${\bf F}_p$, where $$\overline{{\bf F}}_p =\bigcup_{n\geq 1} {\bf F}_{p^n} $$ and ${\bf F}_{p^n} $ is ...
2
votes
1answer
52 views

Does linear dependency have anything to do when determining a span?

Q: Does $\{(1,1) , (2,2)\}$ span $\mathbb{R}^2$? A: No, because they are linearly dependent. I agree that it doesn't span $\mathbb{R}^2$, but from my understanding, linear dependency has ...
2
votes
3answers
104 views

Matrix-free proof of $Z(GL_n(F)) = \{\lambda I:\lambda \in F^\times\}$?

How does one prove that $$Z(GL_n(F)) = \{\lambda I:\lambda \in F^\times\}$$ without resorting to matrices (and bases)? (BTW, $Z(GL_n(F))$ is the center of $GL_n(F)$, the general linear group of order ...
1
vote
1answer
98 views

Example for finite dimensional analog of integral transforms

I understand that integral transforms are generalisations of the dot product of functions that could be interpreted as infinite dimensional vectors. The most significant advantage then is that ...
1
vote
1answer
77 views

Farey Sequence Vector Orthogonality Relation Question

Take the Farey sequence $\mathcal{F}_n$ for $n=39$ with values $a_m\in \mathcal{F}_n$ and put them into a vector $$ \vec v_k=\biggr(\exp(2\pi i k a_m)\biggr)_m $$ Since Merten's function for $n=39$ ...
1
vote
3answers
92 views

Dependency of linear map definition on basis

The definition of linear map depends on the basis. Is this a flaw of the "linear map" construct?