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
votes
2answers
81 views

Decompose $V \otimes V \otimes V$ into irreducible representations of $SL_2(\mathbb{R})$

Let $V=\mathbb{C^2}$ be the standard representation of $SL_2(\mathbb{R})$ Decompose $V \otimes V \otimes V$ into irreducible representations of $SL_2(\mathbb{R})$ I will just consider ...
1
vote
1answer
23 views

Which of the following is a vector subspace of $R^3$

To prove that $F$ is a K-vector subspace of $ E$ it suffices to prove $\alpha f_1+\beta f_2 \in F$ with $(\alpha, \beta)$ $\in K²$ and $f_1 , f_2 \in F$ . For trivial cases and easy ones it seems ...
-1
votes
0answers
25 views

what is the field over “ $K^0$ ”?

What is " $K^0$ " ? And does it have a Basis ? I considered the field over K with the power of zero to be a field with zero dimension so that it cannot have a basis.. is it right or am I wrong?
0
votes
1answer
14 views

Expression related to dual norm on bounded linear functionals

Given a vector space $V$ and a norm $\|\cdot\|$ on $V$, the dual norm $\|\cdot\|^*$ on $V^*$ is given by $\|f\|^* = \sup \left\{\frac{f(v)}{\|v\|}\right\}$ over all nonzero vectors $v$. I've found ...
-2
votes
0answers
26 views

Check if a subset is a subspace of some Vector space

For $V= \{F: \Bbb R \to \Bbb R\}$ and $S=\{f \in V \mid f(2)=f(3)\}$, how do I go about proving $S$ is a subspace of $V$? I think it is but cannot figure out a way to prove it as usual... $f(2) = ...
0
votes
1answer
34 views

Cut Space of Vertices without Orthogonal Complement of Cycle Space?

I am studying sparse graphs where their complements tend to be dense (not sparse). I understand this so that the sparse graph has a sparse adjacency matrix while its graph complement is not most ...
0
votes
0answers
13 views

What is cut space of directed graph (digraph)?

A cut is partition of vertices into two disjoint subsets. Digraph is a directed graph. Cut space is defined for an undirected graph as by Wikipedia where the definition for an undirected graph, ...
4
votes
3answers
222 views

Proof: Sum of dimension of orthogonal complement and vector subspace

Let $V$ be a finite dimensional real vector space with inner product $\langle \, , \rangle$ and let $W$ be a subspace of $V$. The orthogonal complement of $W$ is defined as $$ W^\perp= \left\{ v ...
0
votes
2answers
42 views

Dimension of a vector space of polynomials in 3 variables of degree $d$ over $\mathbb{R}$ [duplicate]

Let $V$ be a vector space of homogeneous polynomials in 3 variables $x_1, x_2$ and $x_3$ over $\mathbb{R}$. What is $\dim V$? I think it will be some expression in terms of $d$ but I am not ...
-1
votes
0answers
30 views

What is the dimension of $\ker f =\{(x^3-x)Q(x):Q \in\mathbb{R}_{n-3}[x]\}$?

I have $$\ker f =\{(x^3-x)Q(x):Q \in\mathbb{R}_{n-3}[x]\}.$$ Here $f$ is the following endomorphism $$f(P) = (x^2-x+1)P(-1)+(x^3-x)P(0)+(x^3+x^2+1)P(1),$$ where $P\in\mathbb{R}_{n}[x]$. My ...
0
votes
2answers
24 views

Basis in the space of polynomials

Do the vectors $\mathbf{p}_1(x)= 2+x+4x^2$, $\mathbf{p}_2(x)= 1-x+3x^2$ and $\mathbf{p}_3(x)= 3+2x+5x^2$ make a basis in the space of polynomials of degree at most $2$? If "yes", expand the ...
0
votes
1answer
29 views

Vector space $V$ , quadratic form $f :V\to R$ . Excercise on rad(F) and a new function.

Let $V$ be a finite vector space and $f:V\to R$ a quadratic form. $F$ is the linear symmetrical form of the quadratic $f$. a) Show that the subset $W = \{ w \in V \mid F(w,v) = 0 \text{ for every } v ...
0
votes
1answer
17 views

Use the Gram-Schmidt procedure to construct orthonormal bases for the subspaces of Rn spanned by the following set of vectors

For part c: How can I quickly tell that the dimension of the subspace is 2? I used the algorithm and got "3" basis vectors before realising that the 3rd one was parallel to one of the others and ...
-2
votes
1answer
29 views

Find the bases of the vector space of terminal sequences [closed]

Let V be the vector space of the sequences $ a = (a_0 , a_1 , a_2 , ...) $ of real numbers who are terminally - finally zero sequences (There is $ N $ such that $ a_n = 0 $ for every $ n > N $ ). ...
0
votes
1answer
17 views

signature of a bilinear form

This question is regarding the proof of a lemma in the book Reflection groups and Coxeter groups by Humphreys section 6.8. Lemma: let $E$ be an n-dimensional real vector space endowed with a ...
0
votes
1answer
25 views

How to determine if (1,0,1,1), (1,1,0,1) , (0,1,1,1) spans $R^4$?

I set up a system where $a(1,0,1,1) + b(1,1,0,1) + c(0,1,1,1) = (1,1,1,1)$ (the standard basis of R4) then i found that $a + b = 1$ $b + c = 1$ $a + b + c = 1$ which implies that $a = c = 0,$ and ...
3
votes
1answer
123 views

How to find the appropriate weights to maximize the third coordinate while the first two are zeros

Let's assume, that $v_1, ..., v_n \in \mathbb{R}^3 $ and $ \lambda_1, ..., \lambda_n \in [0, 1] $ The $ v_1, ..., v_n $ vectors are given. I have to find the appropriate weights ($ \lambda_1, ..., ...
0
votes
1answer
104 views

For these subsets $S$, are they subspace for the indicated vector space $V$

Q1. $V =P_5(R)$ and $S=\{p(x)\mid p(15)=0\}$. I think it is a subspace, but not 100% sure. I tried let $p_1(x)=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+a_5x^5$, such that $p_1(15)=0$ ...
0
votes
0answers
26 views

bilinear form and positive definiteness

Let $B$ a symmetric bilinear form on an $n$ dimensional vector space $E$ with signature $(n-1,1)$. Then there exists a hyperplane $H$ in $E$ in which $B$ is positive definite. How to prove this? Is ...
1
vote
1answer
31 views

If $Ax = O$ has only one solutions, then the columns of A: ${v1, v2…,vn}$ span R?

I've been doing some excersices about inner product and I found something interesting but I don't know if my approach is correct at all. Supose that ${v_{1}, v_{2}, ..., v_{n}}$ is a base for a ...
2
votes
2answers
37 views

Integral of bounded function with limit zero at $\pm \infty$

Very simple question here, I almost feel bad for asking it.. Lets say we have a function bounded between $0$ and $1$. This function is high dimensional: $0<f(X) \le1, ~~~ X \in \mathbb{R}^D$ Now, ...
0
votes
0answers
20 views

Showing that $Im(L^*)=(Ker\: L)^\perp \space \:\mathrm and \:\:Ker(L^*)=(Im\: L)^\perp$

Let $V,W$ be finite-dimensional euclidean or unitarian Spaces and $L: V \to W$ a linear map. I have to show the following: $$Im(L^*)=(Ker\: L)^\perp \space \:\mathrm {and} \:\:Ker(L^*)=(Im\: L)^\perp ...
1
vote
0answers
27 views

Show that $\mathcal{F}$ is a lattice fulfilling Stone's Axiom

Consider $$ f^+(0):=\lim_{r\searrow 0}\frac{f(r)}{r},~~~~~~\mathcal{F}:=\left\{f\in C([0,1],\mathbb{R}): f(0)=0, f^+(0)\text{ exists}\right\}. $$ Moreover, let $\mathcal{F}^+$ be the set of all ...
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 ...
0
votes
1answer
25 views

How to denote dimensions

I am struggling with nomenclature. If I have matrix $M \in \mathbb{R}^2 \times \mathbb{R}^4$ it would be considered an element of an 8-dimensonal vector space. If I index $M$ by two indices $i$ and ...
0
votes
0answers
25 views

Linear Algebra.

I have the exercise: Calculate the coordinates of the point $M = (m_1, m_2, m_3)$, such that $\frac{d(M,P)}{d(M,Q)}=\frac{1}{2}$ where $P = (5, 8, 1)$ and $Q = (4, 2, 2)$, here $d(A,B)$ denotes the ...
0
votes
0answers
30 views

Primordial elements of a vector space

We were given the following problem in our Algebra class. Let $V$ be a $K$-vector space (not necessarily finite dimensional), and fix a basis $(e_i)_{i \in I}$ of $V$. If $x = \sum \xi_ie_i \in V $, ...
0
votes
2answers
72 views

Does there exist a Vector that can't be written as a Tuple of Scalars?

The most abstract/general definition of a vector The most general definition of a vector is as an element of a vector space. Given a vector $u$, we can always say that there exists a vector space $V$ ...
0
votes
2answers
28 views

Set of linear transformations being a vector space

Let $V$ and $W$ be vector spaces, $T$, $T_1$, and $T_2$ linear transformations from $V$ to $W$, and $k$ a scalar. We define new transformations $T_1 + T_2$ and $kT$ by the formulas: $$(T_1 + ...
5
votes
1answer
87 views

Proof of algebraic set involving dimension

I need some help to understand the following proof. Let $k$ a field and $V$ an algebraic set. I note $\mathfrak{m}_P$ the ideal generated by $X_1-a_1,\dots ,X_n-a_n$ in $k[V]=k[X_1,\dots ,X_n]/I(V)$ ...
0
votes
1answer
40 views

Determinant of the matrix representation of an isomorphic linear transformation

Are there any theorems or special properties about the determinant of a matrix representation of an isomorphic linear transformation?
0
votes
1answer
23 views

If a vector subspace is open, then it's the whole space

If $V'$ is a subspace of $V$ and it is open then $V=V'$. I've seen a similar question about this, but they talk about "non-empty interiors", is non-empty intetior similar to talkin about open sets? ...
0
votes
1answer
26 views

I think im missing linear property in this normed vector space how should i approach?

Let $V$ be a normed vector space and $V'$ a subspace, $x\in V-\{0\}$show that a)If $\exists \eta > 0$, such that $\{y\in V :\space\space ||y||<\eta\}\subset V'$ then $\frac{\eta x}{||x||}\in ...
0
votes
0answers
11 views

Affine transformations and its decomposition

Let A and B be affine spaces with respective vector spaces V and W. A map $f$ from A to B is called an affine map if there exists a linear transformation $f'$ from V to W such that ...
0
votes
1answer
23 views

affine spaces, affine hyperplanes [closed]

I am totally confused with the definition of affine spaces and affine hyperplanes. Informally an affine subspace is a space obtained from a vector space by forgetting about the origin. Mathematically ...
0
votes
3answers
44 views

Shortest Distance between planes

This is a question which puzzled our entire math class including our teacher, I'm referring to part (b), we're fine with part (a). We don't understand the reason for taking the dot product and the ...
5
votes
2answers
175 views

Is there a way to cut a an ellipsoid with a plane such that it gives an circle?

I'm trying to answer this In $\Bbb {R^3} $ consider the ellipsoid: $2x^2+3y^2+4z^2=1$ It exists a subspace of dimension 2 which intersection with the ellipsoid is a circle. Justify any answer. ...
1
vote
1answer
18 views

Do automorphisms of infinite-dimensional vector spaces over algebraically closed fields always have eigenvalues?

Let $V$ be a vector space over an algebraically closed field $K$ and let $f:V\to V$ be an automorphism, i.e. a bijective endomorphism. If $V$ is finite-dimensional, we know that the characteristic ...
1
vote
1answer
20 views

When do exterior and tensor algebras commute with dual spaces?

Suppose $V$ is a vector space, and $V^*$ is its dual space. Furthermore, let $\Lambda(V)$ be the exterior algebra of $V$, and let $T(V)$ be the tensor algebra. When do the following two statements ...
4
votes
2answers
73 views

Dimension of the subspace of the polynomial ring over $\mathbb R$

Suppose $P_n =\{ f(x) \in \mathbb R[x] : \deg(f(x)) \leq n\}$ and $W = \{ p(x) \in P_n : p(x) = p(1-x) \}$. Find the dimension of subspace $W$. Firstly I am showing that $W$ is a ...
0
votes
2answers
33 views

Prove this is a metric, what else should I consider?

Let $C_b(\mathbb{R})$ be the space of the bounded continuous functions with values in $\mathbb{C}$ defined in $\mathbb{R}$ ($f:\mathbb{R}\rightarrow\mathbb{C}$) prove that: with $x\in \mathbb{R}, ...
1
vote
1answer
24 views

Finding out vectors that screw up linearly independence when given a set

I want to Find the vector space spanned by $A =$ {$(1,1,0,1),(1,2,-1,1),(3,4,-1,3),(-1,-3,-2,-1)$} By definition it's all the linear combinations I can make with those 4 vectors, However I ...
1
vote
2answers
32 views

Do endomorphisms of infinite-dimensional vector spaces over algebraically closed fields always have eigenvalues?

Let $V$ be a vector space over an algebraically closed field $K$ and let $f:V\to V$ be an endomorphism. If $V$ is finite-dimensional, we know that the characteristic polynomial $\chi_f$ has a zero ...
0
votes
2answers
28 views

Can you give me a hint on this proof of a subspace of vectors?

$V=\{(x_n)\in l^2 | $It has only a finite number of vectors$ \neq 0\}$ prove V is a subspace of $l^2$ but it isn't closed. I have problems understanding what does $l ^2$ means, and what the sentence ...
2
votes
0answers
34 views

Proof $S$ is the intersection of $m$ subspace of $V$ of dimension $n-r$

Let $V$ be a $k$-space with $dim(V)=n$. Let $S\subset V$ be a subspace, $\dim\left(S\right)=k<n $. For each $r\in\mathbb{N}$ with $1\leq r\leq n-k$, prove that $S$ is the intersection of $m$ ...
0
votes
0answers
9 views

What is diemension of R(L) and N(L) and this L is one-0ne , onto?

Let $P_3$ be polynomial of degree at most 3 and $\\L:P_3\rightarrow P_3$ be a linear transformation by $L(p)=xp''(x)-4x^2p'(x)$ Then this linear transformation is one-one , onto ? and what is the ...
1
vote
1answer
27 views

$V$ be a vector space ; $f,g \in \mathcal L (V)$ ; $f\circ g-g \circ f=I$ ; then is the set $\{g^n: n\ge 0\}$ linearly independent?

Let $f,g$ be linear operators on a vector space $V$ such that $f\circ g-g \circ f=I$ , where $I$ is the identity operator on $V$ ; then is it true that the set $\{g^n: n\ge 0\}$ is linearly ...
1
vote
1answer
21 views

Transition matrix of polynomial.

Good night, i need help with this. Find the transition matrix that goes from the basis W to the basis $\left\{ 2,1-2x,x^{2}-1,x^{3}-x^{2}+x\right\} $ I found a basis for W, $\left\{ ...
2
votes
1answer
40 views

Dimension about space of matrices of order 3 over the field of the real.

Consider the vector space of the matrices of order 3 over the field of the real $M_{3}\left(\Re\right)$ numbers. and let S be the subspace such that is spanned by the matrices of the form $AB-BA$. ...
0
votes
1answer
10 views

how to prove non-pivot cloumns is the sum of preceding preceding pivot columns in RREF

For example, we can see that $V_{3}=2V_{1}-3V_{2}, V_{5}=2V_{1}-2V_{2}-V_{4}$, but how can we mathematically prove the theory behind it?