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

0
votes
1answer
94 views

Scale position points in a circle.Look like normal scaling

I have an Art degree, no math involved, so sometimes when doing 3D graphics and envisioning problems, it's hard to search for solutions over the internet since I don't have good pointers for search ...
0
votes
1answer
37 views

How can I find velocity vector at position $P$ of a particle moving in a circular fashion?

My question is: How can I find the velocity vector when the particle is at the point $P$? A particle is going in a circular path around the line $\ell : (x,y,z)= (2+t,-1+2t,3-2t)$. The particle is ...
1
vote
1answer
43 views

$v \times w$ is a bilinear map, antisymmetic and $u \times w =0 \Leftrightarrow $ collinear in tensor product

This is my Attempt for part (b): Let's define: $$\Phi: \mathbb{R}^2 \times \mathbb{R}^2 \longrightarrow \mathbb{R}^2 \otimes \mathbb{R}^2 $$ by the following action: $$\Phi(v \times w) = v \otimes ...
0
votes
0answers
29 views

Momentum with respect to a non-straight axis

For a tetrahedron, we place the mass with the weight vector $(0,0,g)$ in each vertex. The mass center is at $m=\frac{1}{4}(A+B+C+D)$ where ABCD are the vertices. Find $a$ such that that the total ...
2
votes
0answers
51 views

Lattices in $\mathbb{Q}_p^n$ with the same stabilizer

Consider the action of $GL_n(\mathbb{Q}_p)$ on $\mathbb{Q}_p^n$, and let $T$ be the diagonal torus. Let $\Lambda$ and $\Lambda'$ be full-rank sublattices ($\mathbb{Z}_p$-submodules of rank $n$) such ...
1
vote
1answer
38 views

Linear algebra: function and vector space

I'm having problems with these two exercises: 1 - Functions: f(t) = t³ - 1, g(t) = t² + t - 1 and h(t) = t + 2. Is there any K in the real numbers that satisfy this condition: f(t) + k*g(t) = h(t)? ...
0
votes
1answer
241 views

Vector Spaces: difference between Equivalence Classes and Quotient Spaces

I'm reading Herstein's Topics in Algebra and Halmos's Finite Dimensional Vector Spaces. I think I understand that if $V_F$ is a vector space over $F$ and $W$ a subspace of $V$, then there is an ...
2
votes
1answer
728 views

Tetrahedron volume relation to parallelepiped and pyramid

Reading at Mathworld, I came across the subject of tetrahedrons. Particularly calculating the volume with four known vertices. There's a formula which uses the triple product to calculate the volume ...
0
votes
0answers
65 views

Finding the dim of polynomial subspace $S \in P_n$ s.t $f(0) = f(2)$

$P_n$ is the linear space of all real polynomials of degree $\leq$ n, where $n$ is fixed. Let $S$ denote the set of all polynomials $f$ in $P_n$ satisfying a specified condition. Compute the dimension ...
0
votes
1answer
74 views

Prove that V is a vector space

I've been given the following question. My problem is that I'm not really sure what I'm suppose to do. Can someone help me getting started maybe just give me a theorem I could use. Consider the set ...
0
votes
1answer
16 views

Uniqueness quantification of planes in vector space

Say we have two parallel lines $l_1\parallel l_2$. Note: We know the equations for these We know one point on each line, $P\in l_1\land Q\in l_2$. To find the plane $\pi$, such that ...
-1
votes
2answers
69 views

Using Lagrange's Mean Value Theorem to prove equality of norms

I'm looking for a proof using Lagrange's Mean Value Theorem of the following: Prove that in $C^1[0,1]$ vector-space, $\left\|f\right\|=\left|f(0)\right|+\left\|f'\right\|_\infty$ norm is equivalent ...
0
votes
2answers
42 views

Equilateral triangle in a plane [duplicate]

If I have the plane $\pi:x+y-z-1=0$ and the two points $A:(1,1,1), B:(2,1,2)$ as two vertices of an equilateral triangle in the plane $\pi$. How can I find all (I'm assuming only 2?) sets of ...
3
votes
1answer
89 views

Confused about subspaces, how do I picture them geometrically?

I'm a visual learner and I'm having trouble intuitively understanding subspaces. Our professor defined a subspace as a non-empty set closed under linear combinations. However, I'm having trouble ...
0
votes
2answers
45 views

Show that <Lu,u> is greater than or equal to zero

I would like some help with the following problem. Thanks for any help in advance. Let $$Lu(x) = −u''(x)$$ where $u \in ML = C^2(\mathbb{R}) ∩ L_2(\mathbb{R})$. Show that $\langle Lu, u \rangle \ge ...
0
votes
1answer
40 views

Proof for function space

Why is a set of functions $$V = \{ f | f : [a,b] \to \mathbb R \quad a, b \in \mathbb R \} $$ with the addition $$(f+g)(x):= f(x) + g(x)$$ and with the multipilication $$(\lambda f)(x):= f(x) ...
0
votes
0answers
106 views

Vector norm in R^2

I want to create a vector norm in $R^2$ that is not identical to a p-norm. I'm thinking of slightly modifying the $p1$ norm, such as adding a coefficient or exponent. For example, $||x|| = ...
-1
votes
1answer
37 views

Proving that a linear operator is hermitian

I would like some help with the following problem. Thanks for any help in advance. Consider $Lu(x) = iu′(x)$ where $u ∈ ML = C^1(R) ∩ L^2(R)$. Show that $⟨Lf,g⟩ = ⟨f,Lg⟩$ for any $f$ and $g$ from ...
1
vote
1answer
33 views

Relationship between $\hat r$, $\hat \theta$ and the $2\times 2$ rotation matrix

The polar unit basis vector fields are $\hat r(r, \theta) = \cos(\theta)\hat i + \sin(\theta)\hat j$ and $\hat \theta(r, \theta) = -\sin(\theta)\hat i + \cos(\theta)\hat j$. If I have a vector $\vec ...
0
votes
1answer
30 views

Direct sum of 3 vector spaces

I am trying to find some example of 3 subspaces A, B, C of some vector space X, such that $A+B, A+C, B+C$ and $A+B+C$ are all direct sums of X. Is there any ideas? I tries basis vectors but those are ...
3
votes
1answer
29 views

A representing matrix of a linear transformation which acts on a subspace of $\mathbb{R}^3$

$$ \text{Let }V\text{ be a subspace of }\mathbb{R}^3\text{ with dimension 2,} \\ \text{Let }A=\begin{pmatrix} 1 & 2 & 1 \\ 2 & 0 & -1 \\ -3 & 2 & 3 \end{pmatrix} \\ ...
0
votes
2answers
146 views

Determine whether set forms an orthonormal basis

Consider the three vectors $x^{1} = (1/\sqrt2, 0, −1/\sqrt2)^T$ , $x^2 = (0, 1, 0)^T$ , $x^3 = (1/\sqrt2, 0, 1/\sqrt2)^T$. Does the set $A = {x^1, x^2, x^3}$ form an orthonormal basis of ...
0
votes
1answer
37 views

Check if a vector-space is closed or non-closed

Let $V$ denote the subset of $C[-\pi,\pi]$ consisting of all finite linear combinations of functions $1, \cos x, \cos 2x, ... \cos nx, ..., \sin x, \sin 2x, ... \sin n2, ... $ I want to examine if ...
3
votes
4answers
101 views

What are the rules to determine if a set is a basis for V?

So I have here an example, we let $S = \{t^2 + 1, t - 1, 2t + 2\}$ How can I determine if $S$ is a basis for $V = P_2$? Also, do I need to prove that $S$ is linearly independent? I'm almost there, ...
3
votes
2answers
101 views

Can this proof of existence of a Hamel basis using transfinite recursion be shortened/simplified?

This is (I hope) a solution to Problem 112 in A. Shen and N. K. Vereshchagin, Basic Set Theory (AMS 2002). It is - I thought! - a semi-routine exercise, part of whose purpose is to enlighten the ...
1
vote
0answers
33 views

Define a norm using a pseudo norm

Let $V$ be a vector space with a pseudo norm $\rho$ defined on it. Find a subspace $W$ of $V$ such that $X=V/W$ is a normed space with a norm $\|\cdot \|$ Find a linear map $T:V\to X$ such that ...
-4
votes
3answers
522 views

How Can I find partial derivative of integral function with respect to x?

I am given $$f(x,y) = \int_{0}^{\sqrt{xy}} e^{-t^{2}} dt$$ For $x, y > 0$ How can I find partial derivative of $f$ with respect to $x$ ? I am trying to integrate it first but it is something ...
1
vote
1answer
42 views

Algebra of dimensions of Subspaces.

$\newcommand{\span}{\operatorname{span}}$Prove that the following is true for any subspaces $V, W$ of $\mathbb{F}_n$: $$\dim(V + W) = \dim(V) + \dim(W) - \dim(V\cap W).$$ My attempt: Let $V, W$ and ...
1
vote
3answers
676 views

Finding the orthogonal projection of a vector on a subspace

The title could include "subpace with more than one base vector", because that's what I'm having trouble with. Say we have our subspace that is spanned by $\{(1, 0 ,1),(1, 1, -1)\} $ and we have the ...
1
vote
1answer
57 views

Useful definitions of vector addition, dot product and scalar multiplication for strings?

A string is something like: "ad939-0x!", or "mary had a LittLE lambD". There's a character set, and you glue stuff together from the character set to build strings. Have there been any useful ...
3
votes
1answer
36 views

Why is $\left(W_1\cap W_2\right)+W_1'+W_2'=\left(W_1\cap W_2\right)\oplus W_1'\oplus W_2'$?

Let $V$ be a vector space over a field $K$. Let $W_1,W_2,W_1',W_2'$ be subspaces of $V$ such that $W_1=\left(W_1\cap W_2\right)\oplus W_1'$ and $W_2=\left(W_1\cap W_2\right)\oplus W_2'$, then why ...
0
votes
2answers
35 views

Example for a basis of $\mathbb{R}^4$ which consists of several conditions

Let V be a vector space over $\mathbb{R}$ and let $B=\{v_1,v_2,v_3,v_4\}$ be its basis. Let $T : V \to V$ be a linear transformation which satisfies the following condition: $T(v_1) = T(v_2)$ ...
0
votes
1answer
32 views

Question involving Linear Transformations

Two linear transformations $T_1:\mathbb{R}^4\rightarrow\mathbb{R}^4$ and $T_2:\mathbb{R}^4\rightarrow\mathbb{R}^4$ are represented by the matrices $\mathbf{M}_1$ and $\mathbf{M}_2$ respectively, ...
1
vote
1answer
33 views

Inverse of the Wedge of a Matrix

Let $V$ be an $n$-dimensional vector space. Then in the usual way define $\wedge^2 V$ to be the vector space spanned by the elements $v_1 \wedge v_2$ where $v_1, v_2 \in V$ such that they satisfy the ...
0
votes
1answer
37 views

Does a vector space with dimension 1 have an orthogonal basis?

Normally an orthogonal basis of a finite vector space is referred as a basis that contains many vectors, i.e. 2 or more. Consider a vector space that its dimension is 1 - does it have an orthogonal ...
0
votes
1answer
442 views

Proof of subspaces of odd and even functions

$F^+(\mathbb{R})$, the set of even functions in $F(\mathbb{R}, \mathbb{R})=\{ f: \mathbb{R} \to \mathbb{R} \}$ and $F^−(\mathbb{R})$, the set of odd functions in $F(\mathbb{R}, \mathbb{R})$ are both ...
0
votes
2answers
45 views

Proof of subspaces and vector spacecs

Which of the following sets W are subspaces of the given vector space V over the field F? (a) V = R^3, F = R , W = {(a, b, c) ∈ R^3|a^2 + b^2 = c^2} (b) V = m x n matrix, F = R ,W = {AB|A ∈ m x k ...
3
votes
2answers
36 views

For positive operators $A$ and $B$ with $A^6=B^6$ show that $A=B$

Since $A$ and $B$ are positive, I managed to show that $A^6$ and $B^6$ are positive. Now, I can use the fact that there exists a unique square root of both of those and since they're equal, their ...
1
vote
1answer
33 views

Error in my exercise concerning Riesz spaces and Yosida's Lemma.

I was given this exercise in class today: Using Yosida's Lemma, proof that there exists a Riesz-homomorphism $\varphi: BC(\mathbb{R}) \to \mathbb{R}$ such that $\varphi(\textbf{1}) = 1$ and ...
0
votes
0answers
24 views

Multiplicity of Ritz eigenvalues

Consider a Krylov subspace $K_m=span\{v,Pv,...,P^{m-1}v\}$, for $P$ a square matrix and a nonzero vector $v$. Let $H_m$ represent the projection of $P$ (seen as an application) restricted to $K_m$, ...
0
votes
1answer
36 views

What is the dimension of $f(A)$ if $f:A\subseteq\mathbb{R}^k\to\mathbb{R}^n$ is linear and $A$ is a subspace of $\mathbb{R}^k$?

Let $k\le n$ and $f:A\subseteq\mathbb{R}^k\to\mathbb{R}^n$. Obviously, $f(A)$ is a subspace of $\mathbb{R}^n$ iff $\forall x,y\in A:\exists z\in A:f(z)=f(x)+f(y)$ $\forall x\in ...
0
votes
1answer
93 views

Are the domain and range of a vector field vector quantities?

I have been given the following question: "Is this statement true?: A vector field is a function where the domain and the range are vector quantities." I'm unsure of the answer; as the values on ...
1
vote
1answer
331 views

Creating a 3D Plane using the normal and point vector

I'm not understanding the relationship of a normal vector and a position vector that makes it into a 3D plane, and how I can visualize what that 3D plane is going to look like in 3D space. Say I ...
2
votes
2answers
59 views

Uncountable Subset of Reals Generates Reals by Finite Integral Linear Combinations

A question that I thought of earlier today that I couldn't quite get anywhere with. Given an uncountable subset of the reals, $S$, is it always possible for any $r \in \mathbb{R}$ that we can take a ...
2
votes
0answers
66 views

Jordan decomposition algorithm

I'm trying to calculate the value of a matrix function. As far as I understood, this is done by first decomposing my matrix $A$ into $PJP^{-1}$. Where $J$ is in Jordan normal form. However, this ...
0
votes
2answers
32 views

Finding a line in a given a parallelity with a plane, and intersection with another known line

I'm looking for the line $l_1$ which intersects with the line $$l_2:\begin{cases} x = 1 &+& t \\ y = 2 &-& 2t \\ z = -1 &+& t \end{cases}$$ and is also parallel with the ...
1
vote
1answer
28 views

doubt regarding basis of vector space

i am studying basis of a vector space .let vector space be $V$ and subset be $B$. in it the two condition stated were that $B$ is a maximal linearly independent set in $V$ and second condition was ...
1
vote
1answer
24 views

Help with this demostration of spaces

Let U, W, S dimensional subspaces of a finite space such that V=U+W+S. Prove that V = U⊕W⊕S if and only if. dim (V) = dim (U) + dim (W) + dim (S)
1
vote
2answers
236 views

Help with Linear Algebra proof that an infinite set of polynomials is independent

Let $\mu_k(t) = t^k$, for $k = 0,1,2, \dots,$ and $t$ is real. I want to show that the infinite set $S = \{\mu_0, \mu_1 , dots \}$ is independent. To do this, set up (1) $\sum_{k=0}^n c_kt^k= 0$, for ...
1
vote
2answers
2k views

Prove all 8 axioms of a vector space?

$\newcommand{\u}{{\bf u}} \newcommand{\v}{{\bf v}} \newcommand{\w}{{\bf w}} \newcommand{\V}{{\bf V}} \newcommand{\L}{{\bf L}} \newcommand{\W}{{\bf W}} $ I suppose that this question has been asked ...