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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5
votes
0answers
133 views

Eigenvalues of a difference of non commuting products

Let $A$ be an $n \times n$ complex matrix and let $T(M) = AM-MA$. I need to determine the eigenvalues of $T$ in terms of those of $A$. This was an exercise from Artin, and I was not being able to ...
2
votes
0answers
19 views

Question regarding example of toric variety and generators of cone

Consider the canonical example taking n=2, and taking the cone $\sigma$ generated by the vectors $e_{2}$ and $2e_{1} - e_{2}$. The dual cone $\sigma^{v}$ is defined as the set of vectors in the dual ...
1
vote
0answers
24 views

A question on vector components with respect to an embedding on a manifold

I've been working my way through Nakahara's book "Geometry, topology and physics" and I'm currently reading through chapter 7 (on Riemannian geometry). In the 2nd section of the chapter he gives a ...
2
votes
1answer
22 views

Denote $Z$ as the set of points in $\mathbb{R}^n$ whose coordinates are $0$ or $1$. Find the maximum, of the number of points in $Z \cap V$.

Denote $Z$ as the set of points in $\mathbb{R}^n$ whose coordinates are $0$ or $1$. Let $k$ be given, $0 \leq k \leq n$. Find the maximum, over all vector subspaces $V \subset \mathbb{R}^n$ of ...
1
vote
0answers
52 views

Can someone explain to me the mathematics behind this code which calculates the “minimum cosine distance”?

Suppose we have two equal sized arrays of vectors $$ array_1 = \{ (x_1, y_1), (x_2, y_2), ... , (x_n, y_n) \}\\ array_2 = \{ (x_1', y_1'), (x_2', y_2'), ... , (x_n', y_n') \} $$ What the code does is ...
-2
votes
3answers
53 views

Proof in inner product [closed]

I am learning for my finale and having issues proving the two statements: $$ \|U\|=∥V∥ \iff\langle U+V,U-V\rangle=0 $$ $$ \|U\|^2=\|V\|^2=\langle U,V\rangle \rightarrow U=V $$ your help will be ...
6
votes
1answer
73 views

About $\mathbb{R}$ as a vector space over $\mathbb{Q}$

I want to understand better the structure of the vector space $\mathbb{R}$ over $\mathbb{Q}$. I know that it is an infinite dimensional vector space with a non countable Hamel basis, and it is cited ...
1
vote
0answers
20 views

Is $\mathbb{C}[G]$ dual to $U(\mathfrak{g})$?

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. Is $\mathbb{C}[G]$ dual to $U(\mathfrak{g})$? In the case of $G = SL_2$, we have $\mathbb{C}[SL_2] = \langle a,b,c,d\rangle / (ad-bc-1 )$ and ...
1
vote
2answers
40 views

Cauchy sequence which does not converge example.

Consider the normed space $(X, \Vert \cdot\Vert) $ where $$ X=\{ (a_n)_n \quad|\quad (a_n)_n \text{ real sequence with } \lim_{n\to \infty}a_n=0 \} $$ and $$\Vert (a_n)_n\Vert:= \sum_{n\geq ...
0
votes
0answers
44 views

Equation of a hyperplane tangent to a level set

Can anyone help? The normal equation of a hyperplane is But I am unsure how you would specify that it is tangent to the point $(x_0, F(x_0))$ or to the level set.
0
votes
0answers
34 views

Linear Algebra -Subspaces

Are the following subsets of $\Bbb R ^n$ subspaces? Justify the answer. a) The set of all vectors $(x_1, x_2, \dots, x_n)$ for which $x_1x_2x_3...x_n = 0$. For this one I worked it out and got that ...
0
votes
1answer
51 views

Linear Independence of Pre-Image

If $t:V \to W$ is a linear transformation from vector space V to W and A is a subset of V then it seems that if $t(A)$ is linearly independent in W then A is linearly independent in V. See for example ...
-1
votes
1answer
45 views

Find the equation of a budget hyperplane in R4, from an endowment point and a price vector [closed]

http://imgur.com/ofDS6MO Its been a while since I had to deal with vectors, if someone could help me along with 1.1 that would be very much appreciated
1
vote
1answer
44 views

Find $L\left( \left[\begin{matrix} 3 \\ -1\end{matrix} \right]\right)$?

I have the following linear map: $L: \mathbb{R}^2 \rightarrow \mathbb{R}^2$. Now, suppose $L\left( \left[\begin{matrix} 1 \\ 1\end{matrix} \right]\right) = \left[\begin{matrix} 1 \\ 4\end{matrix} ...
6
votes
1answer
53 views

Given a vector space with two inner products, there is a linear transformation taking one to another

I am looking for some hint to the following question: Let $V$ be an $n$-dimensional real inner product space and let $\langle x,y\rangle$ and $[x,y] $ both be two different inner products on V. ...
2
votes
1answer
44 views

What is the tensor product with zero factors?

I am trying to understand what tensor products are, but one particular thing confuses me slightly. For $V$ a vector space over a field $\mathbb{k}$, why do we have that $V \otimes \mathbb{k} \cong V$? ...
1
vote
2answers
63 views

Defining the $L^2$ norm of a vector valued function

I am considering a collection of function of the type, $ f:[0,2\pi]\rightarrow \mathbb{R^2}$. I want to define the $L^2$ norm of the function in that space. I am defining the a norm of ...
0
votes
1answer
32 views

Find a vector equation $r(t)$ for the line through point $(-9,-2,5)$ which is normal to the surface at $(-9,-2,5)$

Can someone help me on this? I don't know which equations to use. I don't understand what they mean by "surface".
1
vote
1answer
38 views

how to understand this identity about the range in linear algebra

I see the identity in page 48 of paper http://arxiv.org/pdf/0909.4061.pdf. Specially, if $U^TU=UU^T=I$, then we will have $U^T\text{range}(M)=\text{range}(U^TM)$, where $\text{range}(M)$ means the ...
0
votes
1answer
18 views

Why the following $dim(N(A)) + dim(C(A^T)) = n$ is true?

I was studying about row, column, null and left null spaces, and one thing I don't understand is why the dimension of the null space of a matrix $+$ the dimension of a the row space is equal to the ...
0
votes
0answers
18 views

On bilinear forms: if $B$ is non-degenerate, then $(U_1\cap U_2)^{\perp_L} = U_1^{\perp_L} + U_2^{\perp_L}$

I think I have a proof, but I'm not completely certain it is correct: Because $B$ is non-degenerate, it follows that if $W \leq V$, then $W = W^{\perp_L \perp_R}$. It is evident, too, that $(U_1 + ...
2
votes
1answer
54 views

Understanding Norms on Vector Spaces

Let $\|\cdot\|$ be a norm (not necessarily the standard norm) on $\mathbf R^2$ and $S$ be the set of all the vectors $v$ such that $\|v\|=1$. For any point $p\in S$, let $\ell_p$ denote the line ...
2
votes
5answers
216 views

If $n=\dim(V)$ and $n$ vectors are linearly independent, then they form a basis

If $V$ is a vector space and $v_1, v_2, . . . , v_n \in V$ span $V$, and $u_1, u_2, . . . , u_m ∈ V$ are linearly independent, then $m\le n$. Use this to prove that if $V$ has dimension $n$ and $u_1, ...
2
votes
1answer
13 views

Dimension of set of all homogeneous polynomial of degree $d$ in $n$-variables over a field $F$

Let, $V$ be a set of all homogeneous polynomial of degree $d$ in $n$-variables over a field $F$. Then dimension of $V_F$ is (A) $\left(\begin{matrix}n\\d\end{matrix}\right)$ (B) ...
0
votes
1answer
27 views

How to find normal vector to a given set of vectors?

Given a vector space $V$ with dimension $n$, if I am given $m<n$ linearly independent vectors $a_1,a_2...,a_m$, will there always be a vector $v \in V$ which orthogonal/normal to all the given $m$ ...
0
votes
0answers
17 views

Projecting/Mapping 2D vector graphics onto 3D-models (like STL-files)

At my working place, I have the task to map or project 2D-vector graphics onto 3D-models (like stickers). This is not at all about rendering, but I would need the exact 3-dimensional coordinates of ...
0
votes
1answer
26 views

determining a linear isomorphism so that two quadratic forms become equivalent

Consider the matrix $$ G = \begin{pmatrix} 3 & 1 & -2 \\ 1 & 2 & 0 \\ -2 & 0 & -3 \\ \end{pmatrix}$$ and the quadratic form $q: \mathbb{R}^3 \to \mathbb{R}$, given by $q(v) ...
0
votes
1answer
26 views

Equations of the same plane

Are \begin{equation*} -x-4y+3z=-9~\text{and}~x+4y-3z=6 \end{equation*} equations of the same plane? I graphed them and they look the same, but I am not sure. Thanks
2
votes
1answer
37 views

Prob. 14, Sec. 3.8 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: A Hermitian positive semi-definite form

Let $X$ be a complex vector space, and let the map $h \colon X \times X \to \mathbb{C}$ satisfy the following conditions: For all $x, y, z \in X$ and $\alpha \in \mathbb{C}$, (i) $h(x+y, z) = ...
1
vote
1answer
22 views

Dual Space of an Euclidian Space is also Euclidic with a specific bilinear form

Let $\gamma: V \times V \to K$ be a nondegenerate bilinear form, and let $\overline{\gamma}$ be defined by: $$\overline{\gamma}: V^* \times V^* \to K, \gamma(x, y) = ...
-1
votes
3answers
121 views

A question on linear algebra [closed]

Let $V$ be an $n$-dimensional vector space and $T$ be a linear operator on $V$. Condition 1: there exists $0\neq v\in V$ such that $v, Tv,\ldots, T^{n-1}v$ are linearly independent. Condition 2: ...
3
votes
1answer
32 views

Linear combination of matrices over finite and infinite fields

Let $F\subset K$ be the fields. Let $A_1,\ldots, A_m$ be the $n\times n$ matrices over the field $F$, and $c_1,\ldots,c_m\in K$ such that $c_1A_1+\cdots+c_mA_m$ is invertible. How to prove that for ...
0
votes
1answer
20 views

What are the least number of extremal points on a polytope?

Given a polytope $P$ embedded in $\mathbb{R}^D$. I can't prove the property that any facet of the polytope will have at least $D$ extremal points lying on it. I can see it for the case $D=2$ but not ...
8
votes
1answer
76 views

Proving that every vector space has a norm.

I am trying to prove that every vector space $X$ has a norm. I have some silly questions, but it's better to ask them now instead of later. I think I'm having a bit of trouble getting intuition about ...
3
votes
2answers
52 views

Subspaces dimensions in $\mathbb{R}^7$

if $U$ and $W$ are subspaces of $\mathbb{R}^7$ and $\dim U = \dim W =4$ then in $U \cap W$ there's a vector different then $0$. I think that it's true, am I correct?
2
votes
2answers
19 views

Coordinate vector equation

I have the following bases which are bases of $\mathbb{R}^3$ $$B = ((1,1,1), (0,1,1), (0,0,1))$$ $$C = ((1,2,3), (-1,0,1), (1,0,1))$$ I need to find if this equation is correct $$[(1,2,3)]_B = ...
1
vote
0answers
54 views

Change of basis formula proof

So I know that this involves using the chain rule, but is the following attempt at a proof correct. Let $M$ be an $n$-dimensional manifold and let $(U,\phi)$ and $(V,\psi)$ be two overlapping ...
3
votes
1answer
48 views

question about direct sum of vector fields and preservation under quotient spaces

Hello all I was given this question in linear algebra it is two parts and asks to prove or give a counterexample. We are given a vector space V and a subspace of it W and the quotient map $ \pi : V ...
0
votes
0answers
26 views

How do I adjust a vector by an amount from another vector

I'm being very dense, but I can't work this out. If I have a direction vector, say (0,1,3) and another vector, maybe ...
1
vote
0answers
14 views

Dependence of isomorphism of two vector spaces with Their fields

V and G be two vector spaces over field F and G respectively. Can they be isomorphic if F is not isomorphic to G? What if F is isomorphic to G? I've no idea where to start...Please help
2
votes
2answers
72 views

Why is the following set not a vector space?

I have the following set and I want to know whether it's a vector space or not: $W = \{(x, y, z) ∈ \Bbb R^3 : (x + y)(2y − z) = 0\}$ Now, I understand that if I have a set W and it's a vector ...
1
vote
2answers
27 views

Need help understanding wiki's informal description of an affine space.

The following was taken from https://en.wikipedia.org/wiki/Affine_space: The following characterization may be easier to understand than the usual formal definition: an affine space is what is left ...
-1
votes
1answer
30 views

On dual spaces and inner products

Let V be a vector space over $\mathbb{C}$ equipped with an inner product $\langle\, , \rangle:V\times V\mapsto\mathbb{C}$. I need to prove that any linear function $\phi:V\mapsto\mathbb{C}$ (element ...
0
votes
3answers
39 views

How to find the basis of the following vector space?

I'm trying to find the basis of the following vector space but I can't seem to be able to find it: $W = \{x = (x_1 , x_2 , x_3 , x_4 , x_5 ) ∈ \Bbb R^5 : x_1 − x_3 − x_4 = 0\}$ I understand that ...
2
votes
3answers
62 views

Are the coefficients of a vector according to a basis unique?

If I have a vector space $V$ ( of dimension $n$ ) over real numbers such that $\{v_1,v_2...v_n\}$ is the basis for the space ( not orthogonal ). Then I can write any vector $l$ in this space as ...
1
vote
0answers
19 views

isometric quadratic spaces over a prime field

Let $(V, \gamma)$ be a quadratic space, where $V$ is an $n$-dimensional $\mathbb{Z}/(7)$-vector space and $r = r(\gamma)$ is the rank of the bilinear form. I want to show: either, $(V, \gamma)$ is ...
0
votes
1answer
28 views

Finding the property of a basis

Let $V = P_2 [x]$, the vector space of polynomials of degree at most 2. Given that $\mathcal B \subset V$, I want to find whether the following is a basis, not linearly independent, not spanning, or ...
0
votes
1answer
35 views

Each bilinear form induces a unique bilinear form from the dual space

Let $V$ be a finite dimensional vector space over a field $K$. Let $\gamma: V \times V \to K$ be a nondegenerate bilinear form. I now want to show that there exists one and only one bilinear form ...
0
votes
0answers
14 views

finding equality with subspaces direct sum

assume that $U_1 \cap U = \{0\}$ and $U_2 \cap U = \{0\}$ $U_1 \oplus U = U_2 \oplus U$? I thought that it's correct because I could find a counterexample.
0
votes
3answers
48 views

How to find the basis of the following vector spaces?

I'm trying, in vain, to find the basis of the following vector spaces: (a) $W = \{x = (x_1 , x_2 , x_3 ) ∈ \Bbb R^3 : x_1 − 2x_2 + x_3 = 0, 2x_1 − 3x_2 + x_3 = 0\}$ (b) $W = \{x = (x_1 , x_2 ...