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

3
votes
1answer
20 views

What series of 'hyperpolyhedrons' do exist? Is there an effective way to derive their cross-sections by 3-d subspace?

There are two obvious series of 'hyperpolyhedrons'. 'Hyperoctahedron' with vertices $(\pm1,0...0), (0,\pm1,0,...0)...(0,...0,\pm1)$ and each vertex connected by an edge with each other vertex ...
0
votes
2answers
48 views

Definite integral-dot product

I have an integral equation containing dot product $$\int_{0}^{L} \left(\frac{a}{L}.b(s)\right)\mathrm ds\tag 1$$ Data Given a is a constant vector of size 3 b(s) is a varying vector of size 3 " . ...
0
votes
0answers
20 views

Proving something is a matroid

I am taking a matroid theory class, and I am having trouble understanding an example we did in class: Let $F$ be a field, $E$ a ground set, and $V$ a vector space over $F$. Let $\phi : E ...
2
votes
3answers
48 views

The “Circle” is a Vector Space?

Consider the set of angles $C = [0, \ 2\pi)$ and, for all $x,y \in C$, define the $sum$ operation as the sum modulo $[0, \ 2\pi)$. The identity element of the addition is the angle $0$. The inverse ...
-1
votes
0answers
19 views

Positive definite [on hold]

I need a graphical representation of positive definite from the eigen values of the matrix which can be expressed from the second degree homogenous equation.
0
votes
0answers
42 views

Quaternion expansion

I have a quaternion equation $ \psi(s)=Pe^{\frac{1}{2}k(s)}\tag 1$ Given conditions and data Here P is a constant unit Quaternion defined for 3D rotation matrix as $(p_1,p_2,p_3,p_4) , p_4\in ...
1
vote
1answer
29 views

Positive definite matrix.

How to illustrate the positive definite matrix in vector space by using the eigen values and eigen vectors?
0
votes
2answers
51 views

Show that vectors of the form (a,b,1) do not form a vector space

Show that vectors of the form $(a,b,1)$ do not form a vector space I tried using the vector space axioms to attack the problem but I am not going anywhere with this problem. I do not need a ...
2
votes
1answer
33 views

Vector spaces - Multiplying by zero vector yields zero vector.

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. The following proof is solely based on vector space axioms. Axiom ...
1
vote
1answer
44 views

A subset that is closed under multiplication but not addition? [duplicate]

I can't get my head around subspaces despite having studied on them quite a lot. Here goes: The problem statement, all given variables and data Give an example of a non-empty subset U of R^2 such ...
1
vote
0answers
34 views

Vector spaces - Multiplying by $-1$ yields inverse element of vector addition.

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. The following proof is based on vector space related axioms. Axiom ...
0
votes
1answer
25 views

Subspace of a vector space Definition

If $W$ is a subspace of a finite-dimensional vector space $V$, then: $\dim(W) \leq \dim(V)$. That makes me think about the definition of a subspace. For example, in $\Bbb R^3$, is $\Bbb R^3$ ...
1
vote
0answers
20 views

Cartesian to geodetic conversion of 3D bounding box - How to calculate latitude and longitude from an axis aligned bounding box

I have a geometry with its vertices in cartesian coordinates. These cartesian coordinates are the ECEF(Earth centred earth fixed) coordinates. This geometry is actually present on an ellipsoidal model ...
0
votes
1answer
40 views

Linear transformations on vector spaces

I'm currently reading up on linear transformations of vector spaces which has gotten me somewhat confused. For instance, there are dilation and contraction operators which can operate on vector spaces ...
5
votes
1answer
76 views

Must a normed vector space be over $\mathbb{R}$ or $\mathbb{C}$?

If it must be, why is this so? In the maths courses I have taken normed vector spaces always have been over $\mathbb{R}$ or $\mathbb{C}$, but I don't see that this has to be so. I am asking because I ...
-1
votes
0answers
17 views

Closure of convex set [closed]

Is the closure of a convex set (in a normed vector space) itself convex? I can't think of a counterexample!
0
votes
1answer
17 views

linear map mapping linearly independent sets

Let $V_1,V_2$ be finite-dimensional vector spaces. $f: V_1\to V_2$ be a linear map. Suppose for $x_1,x_2,\ldots, x_k\in V_1$ we get $\{f(x_1),\ldots, f(x_k)\}$ is a linearly independent set. Does it ...
1
vote
0answers
24 views

$U_1\oplus W=V$ and $U_2\oplus W=V$ but $U_1 \neq U_2$ where $U_1$ and $U_2$ are two subspaces of $V$.

I am searching some counterexamples such that $U_1\oplus W=V$ and $U_2\oplus W=V$ but $U_1 \neq U_2$ where $U_1$ and $U_2$ are two subspaces of $V$ and $V$ is a vector space except $\mathbb {R}^2 ...
2
votes
1answer
38 views

Existence of a subspace with a certain property

I am having trouble solving this problem.I have started solving the problem , so far my guesses for the subspace U were the intersection of V and complement of KerT , but i was soon able to come up ...
0
votes
0answers
24 views

Showing the conjugate symmetric property of an inner product when we don't know if our field is $\mathbb{C}$.

The conjugate symmetric property of an inner product states that $\langle{x, y}\rangle = \overline{\langle{y, x}\rangle}$. My question is regarding showing this when we don't necessarily know that our ...
1
vote
2answers
17 views

Dimension of an intersection.

I have a problem that if $U$ and $V$ are two subspaces of $\mathbb{R}^n$ and $\dim(U)>n/2$, $\dim(V)>n/2$, show that $\dim(U\cap V)\geq 1$. Now I know that $$\dim(U\cap ...
0
votes
1answer
39 views

Is $\sin \theta_{xy}\leq \sin \theta_{xz}+\sin\theta_{yz}$, where $\theta_{ab}$ is angle between unit vectors $a$ and $b$?

Suppose $x,y,z\in\mathbb{R}^n$ are unit vectors. The angle between unit vectors $a$ and $b$ is $\theta_{ab}=\arccos(a\cdot b)$ where $a\cdot b$ is the dot-product. Is $\sin \theta_{xy}\leq \sin ...
0
votes
0answers
13 views

Show that U subspace is supplementary to the kernel. How to find values of a b c d using intersection of two matrices.

I already found the kernel to be \begin{pmatrix} -2c&-2d\\c&d \end{pmatrix}. and U is a subspace of a $M_2$ matrix defined by \begin{pmatrix} a&b\\2a&2b \end{pmatrix}. So i have to ...
3
votes
2answers
37 views

Infinite-dimensional space

I've been dealing with an exercise asking to show that the infinite-dimensional space $R^\infty$ of infinite sequences is isomorphic to a proper subspace of itself. At first I thought I had to show ...
0
votes
1answer
39 views

Zeros of quadratic form of vectors

I have a set of vectors defined as $[\mathbf{v}(x)]_n = e^{jn\pi x}; \quad n = 0 ~\text{to}~ (N-1)$ where $\mathbf{v}$ is an $N \times 1$ vector, $j$ is $\sqrt{-1}$, and $-1 \leq x < 1$. For a ...
0
votes
2answers
47 views

Space of matrices that commute with a given matrix

Let $A$ be an $n\times n$ complex matrix, and $C(A)$ be the vector space of all matrices that commute with $A$. I have to determinate if the dimension of $C(A)$ is greater or equal than $n$, or not. ...
1
vote
1answer
25 views

Lemma before open mapping theorem [closed]

Let (X, | |) and (Y, | |) be normed vector spaces and f a linear mapping of X into Y. Prove that f is an open mapping if and only if there exists an s > 0 such that $B_r^Y (0)\subseteq f ...
0
votes
1answer
31 views

Doubt about subspaces being vector space

Whenever i am saying $V$ is a $n$-dimensional vector space, it means it has $n$ basis vectors each with n elements, right. So when i am proving some theorems or relations involving some ...
0
votes
1answer
11 views

Nonhomogeneous Systems of m equations in n unknowns and Solution Spaces.

My book says that solutions sets of nonhomogeneous systems of m equations in n unknowns is NEVER a subspace of R^n. Why? If we look at any two planes intersecting in R3, there may be a line formed. ...
1
vote
0answers
20 views

Acute angle between plane and line

Find the acute angle between: $x-y-3z=5$ and $x=2-t$ $y=2t$ $z=3t-1$ Here is how I proceed. I take the dot-product of the normal of the plane and the directional vector of the line. This gives me ...
2
votes
1answer
17 views

Dot product of any point on plane and its normal

I was trying to find the distance between a point and a 3D line with parametric equations. On the web, I found a video detailling the steps. https://www.youtube.com/watch?v=9wznbg_aKOo At 2:20, the ...
0
votes
0answers
16 views

Distance between points and parametric equations of line.

Find the distance between the line $x=3t-1$, $y = 2-t$, $z=t$, and each of the following points: a) $(0,0,0)$ b $(2,0,-5)$ c) $(2,1,1)$ Here is how I proceeded: Find v of the line: (3,-1,1) Find ...
1
vote
0answers
21 views

Curve and Constant Curvature

I have initial position vector $p_0$, given curve-linear length $1$. It can be parameterized by $s\in[0,1]$. Assume we have the equation to generate the curve from given starting point and constant ...
2
votes
2answers
29 views

Spectra of operators on different spaces

Can the same operator when defined on two different spaces have different spectra? For example and operator defined on $C_0$ and on $\ell_2$?
0
votes
3answers
39 views

Show that a linear matrix transformation is bijective iff A is invertible.

Suppose a linear transformation $T: M_n(K) \rightarrow M_n(K)$ defined by $T(M) = A M$ for $M \in M_n(K)$. Show that it is bijective IFF $A$ is invertible. I was thinking then that I could show ...
1
vote
0answers
15 views

Proving that $b=(e_1,…,en_,f(e_1),…,f(e_n),f^2(e_1),…,f^2(e_n))$ is base of $E$

Suppose that $E=S\oplus \ker{f^2}$, and $f$ an endomorphism, $S$ is the base of $(e_1,...,e_n)$ and $f^3=0_{L(E)}$. How can I prove that ...
1
vote
2answers
28 views

Do polynomials $ P(t)$ of an odd degree have at least one real root belong to $(t-a)Q(t)$?

This is a continuation of a question where ker(T) = (t-a)Q(t) = P(t). Show that {P(t) ∈ R[t] | deg(P(t)) = 3} ⊂ $∪_{a∈R}$ker(T). So the mark scheme says that all polynomials in R[t] of an odd ...
3
votes
0answers
58 views

Vector spaces - Multiplying by zero scalar yields zero vector

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. The following proof is solely based on vector space related axioms. ...
1
vote
0answers
51 views

What is the solution of greatest possible dimension for this decoupled linear subspace problem?

Let $\left\{A_i\right\}$ be a $k$-element set of $n\times n$ Hermitian matrices, and let $P$ be an $n\times n$ rank-$m$ orthogonal projection matrix. We consider the projection of any matrix $A$ onto ...
4
votes
1answer
44 views

Vector spaces - If an addend adds nothing, then the addend is the zero vector.

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. With one exception, the following proof is solely based on vector ...
1
vote
0answers
42 views

Linear algebra proves

Hi does anyone know how to prove or disprove the following statement. If $V$ and $W$ are subspaces of $R^n$ and $v\in V$ and $w \in W$ then $v + w \in V ∪ W$.
0
votes
0answers
10 views

For what values of λ is this family free (independent), spanning and a basis of R[t]≤3

The family of polynomials $F$ = {${(λ^2 − 1)t^3 + t^2, λt^3 + t − λ, (1 − λ)t^3 + t + 1, λ}$} in $R[t]_{≤3}$ I set their sum to 0 to find the values for it to be independent. $a((λ^2 − 1)t^3 + t^2) ...
0
votes
1answer
25 views

Metric for vector sets

I am currently working on a classification algorithm. Each class is represented by a set of 3D vectors. The cardinality differs for each class. The order of the vectors in a set is completly random. ...
1
vote
1answer
50 views

Which axiom makes a vector space flat?

First of all, I'm not sure if this question even makes sense, i.e. is there a notion of curvature on a vector space structure. However, when dealing with vector spaces (here I am mostly thinking of ...
1
vote
0answers
24 views

Another proof of uniqueness of identity element of addition of vector space

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. The following proof is solely based on vector space axioms. Axiom ...
3
votes
5answers
303 views

Proof of uniqueness of identity element of addition of vector space

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. Proof of uniqueness of identity element of addition of vector space ...
1
vote
1answer
35 views

Trouble proving that $\dim{(W1+W2)}=\dim{(W1)}+\dim{(W2)}-\dim{(W1\cap W2)}$

Let me first summarize the part I understand: $\dim{W_1}=n$, $\dim{W_2}=m$, $\dim{(W_1\cap W_2)}=k$ I've defined $C$ as a basis for $W_1\cap W_2$ with $C = \{v_1, ... v_k\}$. I've expanded $C$ ...
0
votes
1answer
51 views

project a point onto the intersection of surfaces

I have several non linear equations $g_i$ that represent surfaces $s_i$. Their intersection form the surface $S$. For example $s_1 : g_1(x_1,x_2,...,x_n)=c_1$ ... $s_n : g_m(x_1,x_2,...,x_n)=c_m$ ...
3
votes
1answer
84 views

Would the author most certainly be talking about a vector space over $\mathbb{R}$/$\mathbb{C}$ here?

I am working on the following problem in Serg Lang's Linear Algebra book: In the vector space of functions, what is the function satisfying the condition VS2? For reference VS2 is: There is ...
1
vote
1answer
32 views

Covariant and contravariant bases on a diffeomorphism

If we allow two domains $\Omega, \bar{\Omega}\in \mathbb{R}^3$, allow $\mathbf{\Theta}: \Omega \to \mathbf{E}^3$ and $\mathbf{\bar \Theta}: \bar \Omega \to \mathbf{E}^3$ to be two ...