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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. In other words, these are the spaces where ...

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30 views

Proving that function space is a vector space over field

I'm reading Serge Lang's (S.L) linear algebra book. In the beginning, at function spaces section there is such a text: Let $S$ be a set and $K$ a field. By a function of $S$ into $K$ we shall ...
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0answers
53 views

Find the correct functor from $Set $ to $Vect $

Let $Set$ be the category of sets, and $FinVect$ the category of finite dimensional vector spaces. Given a functor $F: Set \rightarrow FinVect$ such that for $K \in Set$, $$F[K]=span_\mathbb{Q} \{ \...
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3answers
34 views

Finding vector length for high dimensions

How do I find the vector length for high dimensions?.We can find vector length for 3d with the formula $\sqrt{v_1^2+v_2^2+v_3^2}$ Likewise how to find the vector magnitude for high dimensions?
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1answer
32 views

Determining the dimensions of subspaces of M?

Let $M$ be the space of three by three matrices. Let $A = \begin{bmatrix}1&0&-1\\-1&1&0\\0&-1&1\end{bmatrix}$. Interpreting $A$ as a linear operator from $M$ to $M$ (rather ...
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1answer
28 views

Linear Independence of Elements over Binary Finite Fields

Consider the binary finite field $\mathbb{F}_{2^q}$ for some $q$. consider a set of elements $r_i$, $1\leq i \leq m$, in $\mathbb{F}_{2^q}$ where $m<2^q$. My question: How to check that elements $...
5
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2answers
131 views

On the sets of sums $\sum\limits_{n=1}^\infty\frac{a_n}{n^s}$ with $(a_n)$ periodic and integer valued, for different values of $s$ natural number

For every positive integer $s$, let $A_s$ denote the set of the sums of the converging series $\sum\limits_{n=1}^\infty\frac{a_n}{n^s}$ for every periodic sequence of integers $(a_n)$. Then each $A_s$...
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0answers
25 views

Defining the Lie bracket on a tensor product Lie algebra

So, my question is the following: Suppose that we have two Lie algebras $(\mathfrak{g}_1,[\bullet,\bullet]_1)$ and $(\mathfrak{g}_2,[\bullet,\bullet]_2)$. Then we can define the tensor product of ...
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0answers
27 views

Perimeter of the circle on a complex vector space

Let $V$ be a complex vector space of dimension $1$. Let $||.||$ be a norm on $V$ and let us note $C_R$ the circle of center $0$ and radius $R$ associated to this norm. ($R>0.$) Let $dz$ be a Haar ...
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0answers
22 views

Proof about ordered vector spaces

We have an ordered vector space of N dimensions and ordering is defined by the L1 norm between vectors. We randomly pick K vectors from that space. For every vector i, we order all vectors based on ...
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1answer
29 views

How many Scalers can be built using three different unit vectors?

I have three unit vectors in a problem: $\hat{t}= (cos(t),0,sin(t)),$ $ \hat{m}= (0,0,1),$ $\hat{n}= (sin(th),0,-cos(th)).$ I know the solution for the problem is: $(-sin(2t)+ 5 sin(2t-4th)+2 sin(...
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1answer
41 views

For finite-dimensional vector space $X$, does $\phi: X \rightarrow \mathbb{R}$ continuous and bounded imply $\phi$ linear?

In a finite-dimensional vector space $X$, if $\phi: X \rightarrow \mathbb{R}$ is linear, then it can be shown to be continuous and bounded. Bounded in this context means that there exists an $M < \...
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5answers
65 views

How to construct a matrix given the null basis of A?

Construct a 4x4 matrix A such that ((1,2,3,4),(1,1,2,2)) is a basis of N(A). So i know that A will have two pivot columns and two free columns, but beyond this I'm not sure how to approach/solve.
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0answers
50 views

Determining eigenvalues of a linear transformation on a field extension via embedding into $\Bbb{C}$

I have been working on the following question: Suppose that $K$ is an extension of $\Bbb{Q}$ of degree $n$. Let $\sigma_1,\dots,\sigma_n:K\hookrightarrow\Bbb{C}$ be the distinct embeddings of $K$ ...
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0answers
11 views

Annihilator concept in dual of a vector space in linear algebra…

Let V be a finite dimensional vector space over field F. Let U and W be two subspaces of V; U° and W° be the annihilators of U and V, respectively. Statement : If U is a subset of W, W° is a subset ...
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0answers
73 views
+150

Element which is independent of a basis

Let $V$ be a complex vector space with a pairing satisfying $(x,y)=-(y,x)$ for all $x,y\in V$. Chosse a basis $v_i$ of $V$ and a basis $w_i$ such that $(v_i,w_j)=\delta_{ij}$. Why is the element $\...
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1answer
48 views

show that $\operatorname{rank}(g\circ f) \leq \operatorname{rank}(f)+\operatorname{rank}(g)-n$

Let $E$ a vector space and $\dim(E)=n$ and let $f,g \in L(E)$ show that $\operatorname{rank}(f\circ g) \leq \operatorname{rank}(f)+\operatorname{rank}(g)-n$ I can see that $\operatorname{Ker}(g) \...
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0answers
29 views

Defining tensors as multilinear maps, without defining the dual space first

A $(p,q)$-tensor can be defined as a multilinear function in several ways, which are mostly equivalent: $$T:(V^*)^p\times V^q\to\mathbb R$$ or $$T:V^q\to V^p$$ or $$T:V^q\to \mathbb R\times V^p$$ ...
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2answers
23 views

Annihilator in dual space [closed]

Let U and W are subspaces of a vector space V. If U is subset of W, inh(W) is subset of inh(U). Is the Converse true? How?
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0answers
36 views

Can the “symmetric algebra” over $\mathbb R^n$ be defined from an infinite-dimensional exterior algebra?

https://en.wikipedia.org/wiki/Symmetric_algebra If I understand that article correctly, the symmetric algebra $S(\mathbb R^n)$ is (isomorphic to) the algebra of polynomials with $n$ variables. As a ...
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2answers
39 views

$\text{proj} _{u_1}v + \text{proj} _{u_2}v = \text{proj} _{U}v$ [closed]

Let $U = $span$\{u_1,u_2\}$ , $u_1,u_2$ are orthogonal and $v$ be a vector. Does the following holds true? $$\text{proj} _{u_1}v + \text{proj} _{u_2}v = \text{proj} _{U}v$$
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4answers
32 views

Show that four points given by vectors lay on a circle

I'm stuck on problem 2.10 from Vector Analysis and Cartesian Tensors by Kendall: Show that the four points with position vectors $\vec{r_1}$, $\vec{r_2}$, $\frac{r_2}{r_1}\vec{r_1}$, $\frac{r_1}{...
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1answer
25 views

Simplex in a complex vector space

I feel this may be a stupid question, but when I look up things on convexity, all definitions are in $\mathbb{R}^n$. For example the definition of a simplex or Caratheodory's theorem. I can only find ...
3
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3answers
51 views

Let $W_{1},W_{2}$ be sub-spaces of $\mathbb{R}^{4}$, find a subspace $W_{3}$ s.t $W_3\subset W_{2}$ and $W_{1}\oplus W_{3}=W_{1}+W_{2}$

Let $W_{1},W_{2}$ be linear sub-spaces of $\mathbb{R}^{4}$. $W_{1}=\text{sp}\{(1,2,3,4),(3,4,5,6),(7,8,9,10)\}$ $W_{2}=\text{sp }\{(x,y,z,w)| \ x+y=0\}$ Find a linear subspace of $ \ \mathbb{...
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4answers
45 views

Can the given vector be expressed as a linear combination of the other three

Four vectors from $\mathbb{R}^4$ are given: $$v_1 = (1, -2, 0, 1), v_2 = (0, 1, -1, -3), v_3 = (1, 0, -2, -5), v_4 = (-3, 5, 1, 0)$$ Can the vector $v_4$ be expressed as a linear combination of the ...
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2answers
15 views

Show that given vectors are not vector space basis in $R^4$

The task is to show that the vectors $(1, 0, 0, 0)$, $(1, 1, 0, 0)$ and $(1, 1, 1, 0)$ are not vector space basis in $R^4$. I tried the method of getting a system of linear equations: $\alpha(1, 0, 0, ...
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0answers
20 views

Test for Spanning of non-$\mathbb N\times\mathbb N$ systems using determinants

If we have an $\mathbb N\times\mathbb N$ system for example $\mathbb R^4$: $[1,1,0,0],[1,2,-1,1],[0,0,1,1],[2,1,2,-1]$. We can proceed to use the determinant to see if the vectors span $\mathbb R^4$. ...
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0answers
25 views

How to prove vector space axioms for a 3 tuple vector lying on a plane? [closed]

Prove that which axioms of vector space are not satisfied for X which is set of all 3-tuples (α1,α2,α3) such that 2α1 −α2 +5α3 = 1, together with operations of R3 (R denotes real numbers).
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0answers
70 views

Function spaces as infinite dimensional vector spaces: $\mathbb{R} \rightarrow \mathbb{R}$ [closed]

I am having difficulty understanding function spaces as infinite dimensional vector spaces. I am interested in functions of the type $\mathbb{R} \rightarrow \mathbb{R}$. If we pick an infinite set of ...
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2answers
42 views

Cosine similarity vs angular distance

While checking Google's Universal sentence encoder paper, I found that they mention that using a similarity based on angular distance performs better than raw cosine similarity. More specifically, ...
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2answers
26 views

Orthogonal projection of a point to plane

A point and a plane is given: point $P(-4, -9, -5)$ and plane defined with three points : $A(0, 1, 3)$, $B(-3, 2, 4)$ and $C(4, 1, -2)$. So far I've managed to calculate the equation of this plane $5x ...
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1answer
30 views

Vector spanning a space

Does a matrix such as: $ \begin{pmatrix} a & b\\ c & d\\ e & f \end{pmatrix} $ Which is a $3\times 2$ matrix span the third dimension if possible. I know its homegenious equation as ...
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4answers
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Logic behind the formula for distance between a point and a line

I just watched a YouTube video that used a neat formula, but I don't understand why it works. Question: Find the distance between the point $(4,1,-2)$ and the following line $$\begin{cases} x(...
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1answer
36 views

If $0 \in S_1+S_2$ then $(S_1+S_2)^* \subset S_1^* \cap S_2^*$.

Let $S$ be a set in V (vector space). The following set is called dual of $S$: $$S^* = \{y \in V \mid \langle y,x\rangle \geq 0 \ \ \forall x \in S \}$$ Let $S_1,S_2$ be nonempty subsets of $V$. I ...
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1answer
15 views

Is a vector apart of a certain Null Space?

I have found that for a certain matrix, the basis for Nul(A) is: span((-7,2,1,0),(-4,0,0,1)) The question posed is if (-3,2,1-1)^t (ie. transpose) is apart of Nul(A). I have worked out that the ...
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2answers
24 views

Show that set of all solutions $(a, b, c)$ of the equation $a+b+2c=0$ is a subspace of a vector space $V^3(R)$

I wonder, as should I be able to show that (1) the solution set satisfies all the axioms of a vector space? (2) or, should I be able to show that for the solution set $W=\lbrace (a,b,c): a, b, c \in ...
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1answer
45 views

Why is the set of all Real Upper Triangular Square matrices not a vector space?

My textbook indicates that the set of all upper triangular n ✕ n matrices is a real vector space, but the set of all upper triangular square matrices is not a real vector space. Why is there a ...
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1answer
52 views

Finding rank and signature of a quadratic form.

Let $$ A = \begin{bmatrix} 1 & 2 & 0 \\ 0 & 0 & -2 \\ 0 & 0 & 1 \end{bmatrix} $$ and define for $x,y,z \in R$ , $ Q \begin{bmatrix} x,y,z \end{bmatrix} = \...
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3answers
38 views

Surjectivity of a linear transformation from the space of all formal power series to itself

Today, one of my friends asked that if we take the linear transformation $T : P(\mathbb{R}) \to P(\mathbb{R})$ s.t. $T(p(x))=p(x)+p''(x)$, then $T$ is onto or not. $P(\mathbb{R})$ is the vector space ...
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1answer
41 views

Example of a certain vector space [closed]

Let $k$ be a field. I am looking for an example of infinite dimensional $k$-vector space $V$ such that $V\cong V\oplus V$. Also kindly let me know in brief how you've thought about it . Thanks in ...
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1answer
107 views

is there a tornado-ish equation or vector 3D? [closed]

the formula I've successfully found that $m = 2\ln(x^2+y^2) $ look like a really static and not moving tornado. But in the same time with vector equation I've found how to twist a cylinder. $ r = \...
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2answers
33 views

Finding which sets are subspaces of R3

https://i.stack.imgur.com/Bpl28.png Hello. I have attached an image of the question I am having trouble with. I thought that it was 1,2 and 6 that were subspaces of $\mathbb R^3$. Here is my working:...
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0answers
39 views

Endomorphisms of $\mathbb Z$-modules are the same as those of $\mathbb F_p$-vector spaces

Let $p$ be a prime number. Consider a direct sum of $n$ copies of the cyclic group of order $p$, written $C_p$: $G=C_p^{\oplus n}$. By a comment in this question, given an $R$-module $M$ and an ...
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1answer
62 views

Is this ordered set of vectors a dcpo?

The definition of a dcpo, or directed complete partial order, can be found here. A real vector is formed with a basis set $B =\{ e_1, e_2 \ldots \}$, and the real numbers $\mathbb{R}$. A vector is a ...
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1answer
31 views

Why is a space of functions equal to the n-fold product of field F?

The space of functions whose domain is $S$ and their values lie in $F$ (a field) is shown by: $$\mathbf{Func}(S,F) = \{f: S \to F\} (1)$$ In linear algebra by Peter Petersen on page 11 example 1.4....
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0answers
27 views

Orthogonality relationship with the kernel

I'm struggling with understanding, visually, the determinants of spaces that are orthogonal to the kernel of a given matrix. My TA gave me the hint to think in terms of the standard dot product? ...
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0answers
24 views

About isometry of direct sum

Let's say we have direct sum of $W$ and $W'$ vector space as product of reflection say map $\pi$. If we restrict this map to $W$ then still it is reflection or not? I think it is reflection since if ...
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1answer
32 views

Direct Sum Test for more than 2 subspaces

I know that if 2 subspace $U_1,U_2$ of V ,can be written as direct sum iff 1) $U_1+U_2$=V 2) $U_1\cap U_2={0}$ I can prove this .But Now to extend this defination to more than 2 subspaces then there ...
2
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1answer
41 views

$\operatorname{rank}(T^3) + \operatorname{rank}(T) \geq 2 \cdot \operatorname{rank}(T^2)$ for finite-dimensional $V$

Consider a finite-dimensional vector space $V$ over the field $\mathbb{F}$ with a linear operator $T.$ Prove that we have $\operatorname{rank}(T^3) + \operatorname{rank}(T) \geq 2 \cdot \operatorname{...
2
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3answers
53 views

One-forms are dual to tangent vectors

In my class it was said that "A tangent vector $X \in T_p(\mathbb{R}^n)$ acts on a one-form to give a real number" and "A one-form acts on a tangent vector to give a real number" Now the '...
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2answers
30 views

Given 3 subspaces, Find dimension of intersection of subspaces

If three subspaces in $R^3$ $w_1 = \{ (x,y,z): x+y-z=0\} , w_2 = \{ (x,y,z): 3x+y-2z=0\}, w_3 = \{ (x,y,z): x-7y+3z=0\} $ then find $dim(w_1 \cap w_2 \cap w_3), dim(w_1+w_2)$ My attempt : basis of ...