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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orbital behavior of objects in a space-time contraction field

I would like to find the orbital velocities of non-gravitating objects imbedded in a space-time contraction field. The field has the form as shown in figure 1. The surface space-time compression ...
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1answer
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$A(2,1,3),B(3,2,4),C(1,8,9), D(4,3,12).$ Find the volume of a parallelepiped with vectors $\vec{AB}$, $\vec{AC}$, $\vec{AD}$.

$A(2,1,3)$ $B(3,2,4)$ $C(1,8,9)$ $ D(4,3,12)$ Find the volume of a parallelepiped with vectors $\vec{AB}$, $\vec{AC}$, $\vec{AD}$. I am not sure how to calculate this. How do I calculate the ...
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1answer
21 views

Kryszeg's Functional Analysis Section 2.8: How is the canonical embedding map injective?

Let $X$ be a vector space over the field $K$ of the real or complex numbers. Let $X^*$ denote the vector space of all linear functionals defined on $X$, and let $X^{**}$ denote the vector space of all ...
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2answers
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Equal-area spherical shell partitioning

I'm trying to solve a particular problem that arose in a computer graphics context, but can be generalised to a bigger problem as well. I'm not entirely sure if this question belongs to MathExchange ...
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34 views

Vector space of dim n and its subspaces

Let $V$ be a vector space of dim $n$ over a finite field $F$ with $q$ elements. (a) Find the no.of dim 1 subspaces of $V$ (b) For each $1\leq k \leq n$, find the no.of dim $k$ subspaces of $V$ My ...
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2answers
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If $K \leq L$ a finite extension then it is algebraic.

I am looking at the proof of If $K \leq L$ a finite extension then it is algebraic. The proof is the following: Let $[L:K]=n<\infty$. Let $a \in L$. We will show that $\exists$ a ...
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Decomposition of order-$n$ tensors

If $V$ is a finite-dimensional vector space, then $V\otimes V\cong\mathbf{S}^2(V)\oplus\bigwedge^2(V)$. The first summand on the right is the symmetric part of $V\otimes V$ and the second summand is ...
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Understanding linearly independent vectors modulo $W$

We've learned in class: Let $W \subseteq V$, a subspace. $v_1, \ldots, v_k \in V$ are said to be linearly independent modulo $W$ if for all $\alpha_1, \ldots, \alpha_k: \sum_{k=1}^n \alpha_i v_i ...
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1answer
54 views

Spaces of polynomials

Let $p_{1}(x)=-2x+2$, $p_{2}(x)=x+2$, $p_{3}(x)=x^{2}+2x+3$, $p_{4}(x)=x^{2}-x+3$. a) From the above four polynomials, determine a linearly independent subset that spans the polynomials. ...
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1answer
27 views

Understanding normal and binormal of a vector or of a spline

I found a paper where it computes the 3D trajectory of a quadrotor and defines an error position as the difference between 2 vectors (here the source, under 3D trajectory control): $$ e_{p} = ...
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3answers
48 views

Whether the set of functions $(1,e^{x},e^{-x})$ linearly independent

Are the set of functions $(1,e^{x},e^{-x})$ linearly independent? I wrote it as an augmented matrix but it brought me to nowhere. Can somebody help me?
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4answers
65 views

How does $\dim \mathbb C$ work?

In the Wikipedia page about Dimension (vector space), it says the dimension of complex numbers is 2 or 1 if it's complex or real vector space respectively. How does that work? How to I describe ...
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26 views

If [x,z] = 0 $\implies$ [x,y] = 0, then y = $\alpha$z. True for infinite dimensional vector space?

I'm reading Halmos's Finite Dimensional Vector Spaces, in which he makes several references to the infinite dimensional case. In my edition this item appears as question 6 at the end of section 14. ...
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2answers
36 views

Finding the basis and dimension of a vector space

Find the basis and dimension of vector space $ L_{1}$ spanned by vectors $ a_{1} ,a_{2},a_{3} $, the basis and dimension of vector space $ L_{2}$ spanned by vectors $ b_{1} ,b_{2},b_{3} $ and also ...
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1answer
45 views

Symmetric algebra

If $V$ is a vector space over the field $K$ with basis ${v_1, v_2,…,v_n}$, then the symmetric algebra $S(V)= K[v_1,v_2,..,v_n]$. The question is: If $K$ is a commutative ring, then this equality is ...
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2answers
64 views

$V \cong V \oplus V$ as $K$ vector spaces

I am not very sure about the triviality of this problem but I can't see the solution. Problem is If $V$ is a countable dimensional vector space over field $K$, then as $K$ vector spaces $V \cong V ...
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2answers
312 views

Weird isomorphisms of infinite groups

According to my interpretation to one of the answers in Splitting in Short exact sequence, $$\Bbb R \cong \Bbb Q \oplus \Bbb R / \Bbb Q$$ also, according to What is known about the quotient group ...
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1answer
32 views

stuff involving adjoint, self adjoint [on hold]

Let $T: V \to V$ be a linear transformation relative to a finite dimensional Euclidean space $V$ (real or complex). Prove that there exists linear transformation $T^*: V \to V$ (called the adjoint ...
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1answer
38 views

Calculating a spread of $m$ vectors in an $n$-dimensional space

My question is regarding spreading $m$ vectors in an $n$ dimensional space such that the vectors are maximally distant from each other. For example, let us say I have a 2-D space, and 3 vectors, the ...
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1answer
19 views

Inner product respect on a non-canonical base

Let a,b be vectors, on the standard base we use the dot product by simply doing a.b. But when we consider an other base we put a symmetric matrix between them. Why? How does that work? Thanks
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1answer
24 views

Vector cross-multiplication [closed]

Given v = (a b c), w = (x y z), what are the components of the cross product of v and w?
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3answers
23 views

How much should I scale $dx$ and $dy$ individually to get a vector of required magnitude

I have a $dx$ and a $dy$ and I need to create a vector of magnitude $35.5$ in that $(dx, dy)$ direction. How much should I scale $dx$ and $dy$?
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1answer
37 views

Determining a basis for a space of polynomials

Determine a basis from the following set of second degree polynomials. Does this basis span the space of the second degree polynomials? What is the dimension of the (sub)space that it spans? ...
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40 views

Whether a set of vectors span a subspace that includes a given vector

Do the vectors $(0, 1, 2), (1, 2, 1), ( -1, 2, 4)$ a) span $\mathbb R^{3}$ b) span a subspace that includes $w = (-2, 2, 10)$ I know they don't span $\mathbb R^3$ since they are ...
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3answers
32 views

Property of eigenvectors in linear mapping

Let $V$ be a bector space over a filed $\mathbb{F}$, and let $L:V\rightarrow V$ be a linear mapping. Let $U$ be a subspace of $V$ such that $L(U)\subset U$ Suppose that $u$ and $v$ are eigenvectors ...
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1answer
30 views

Prove dimension of sum of two subspaces

Let $U$ and $W$ be subspaces of $\mathbb{R^n}$ where $\dim(U)=n-1$, $\dim(W)=n-3$ and $n\geq 3$ Prove that $\dim(U\cap W)\geq n-3$ I used the property that both $U$ and $W$ are subspaces of ...
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2answers
38 views

How to tell if two spherical coordinates lie on the same plane

I have the rho, theta, and phi values of two points, how can one tell that two vectors are normal to the same plane by looking at their spherical coordinates?
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5answers
180 views

Proof involving subspaces

I encountered this question in a document I found on a google search, it bugged me because my perception keeps telling me I'm wrong no matter what I do. Let $U$, $W$ and $Z$ be subspaces of a ...
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1answer
17 views

showing a basis is suitable for a dual space, from linear forms

Let us define $B = b_1, b_2, b_3$; where $ b_1(f) = f(0)$, $ b_2(f)=−f'(0) $ and $b_3(f) = f''(0) $. Let $E ^∗$ be the dual basis of $E = {1, x, x^2}$. Show that B is a basis of the dual ...
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28 views

Arrow Space Construction

Is there a paper or book that has rigorously constructed the space of "arrow vectors" and shown that it is a vector space? I'm just wondering how far anyone has followed the heuristic.
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1answer
70 views

Steinitz's Lemma - Removing

In the book that I am using, Linear Algebra Done Right, the proof for the Steinitz exchange lemma (which can be found here) left me unconvinced. The proof refers to the linear independence lemma. ...
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1answer
24 views

Space is direct sum of subspaces - propostion conditions giving me problems

In Sheldon Axler's "Linear Algebra Done Right" - 2$^{\textrm{nd}}$ Edition, on the section for Direct Sums, the following proposition is stated. Following this is the proof of this 'if and only ...
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1answer
36 views

computation involving exterior $2$-form on $\mathbb{R}^n$

Let $$\theta = \sum_{i=1}^{n-1} x_i \wedge x_{i+1}$$be an exterior $2$-form on $\mathbb{R}^n$, and $A, B \in \mathbb{R}^n$ are vectors$$A = (1, 1, 1, \dots, 1),\text{ }B = (-1, 1, -1, \dots, ...
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2answers
201 views

Is it possible that there isn't a linear span which precisely spans a vector space?

In the assignment I'm asked to decide whether given: $S = \Bigg \{ \begin{bmatrix}a &b \\ c &d\end{bmatrix} \in M_2(\mathbb{R}) \; | \; ad = 0 \Bigg \},\mathbb{F} = \mathbb{R}$. $S$ is a ...
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3answers
35 views

What ring-sum of vector spaces can possibly mean?

I'm given this test assignment, and I can't decipher what it says. Would you kindly help me? Here's the assignment itself: Let $U$ and $W$ be sub-spaces of the linear vector space $V$ s.t. $U ...
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6answers
557 views

Is there such thing as an unnormed vector space?

I learned about Banach spaces a few weeks ago. A Banach space is a complete normed vector space. This of course made me wonder: are there unnormed vector spaces? If there are, can anyone please ...
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2answers
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Vectors: $a = (1,2)$, $ b= (2,-1)$, $c = (-5, 20)$ Find values for $k$ and $l$ for $c = la + kb$

We have these vectors: $a = (1,2)$, $ b= (2,-1)$, $c = (-5, 20)$ and I have to find values for $k$ and $l$ given this: $c = la + kb$ How do I go on about solving this one? Do I have to calculate ...
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1answer
64 views

Vector Spaces: Tensor Product

Reference Foundation for: Hilbert Spaces: Tensor Product Problem Given a vector spaces $V$ and $W$. Take its algebraic tensor product: $\tau:V\times W\to V\otimes W$ How to prove that the image ...
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3answers
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To find the two dimensional subspace of $R^{3}$

I am stuck with this question .Kindly help me to get through this Option A is of 1 dimension so it cannot be answer but all other options are looking fine to me , What i am missing ? THANKS
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1answer
21 views

Null/Col/Row space be a line\plane through the origin?

For a $4\times3$ matrix can the nullspace, the column space and row space all be lines through the origin? For a $2\times4$ matrix can the nullspace, the column space and row space all be planes ...
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1answer
79 views

Does every isomorphism between $V$ and $V^*$ send some basis to its dual basis?

Suppose that I have a vector space isomorphism $\theta: V \to V^*$ where $V$ is any vector space (probably over $\mathbb{C}$ is required) and $V^*$ is its dual space. Is it always possible to find a ...
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282 views

Every invertible linear transformation can be perturbed a bit without destroying invertbility, Neumann series

Let $T: V \to V$ be any linear transformation on a real or complex vector space $V$. Show that there exists $\epsilon_0 > 0$ $($depending on $T$$)$ so that $I + \epsilon T$ is invertible for any ...
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Direct sum of two spaces

Let $\alpha_1=[1,1,0,1]$, $\alpha_2=[1,0,1,1], \alpha_3=[1,1,1,1],\alpha_4=[0,1,1,1]$ be a vectors from $\mathbb{R}^4$ let $U=span(\alpha_1, \alpha_2) \ and \ W=span(\alpha_3, \alpha_4)$ Check that ...
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Reference for work on abelian divisible groups $G$ such that for every $n \in \mathbb N , g \in G , \exists$ unique $x \in G$ such that $g=x^n$

Is there any work or reference in the literature about those abelian divisible groups $G$ such that for every $n \in \mathbb N , g \in G , \exists$ unique $x \in G$ such that $g=x^n$ ; I think then I ...
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2answers
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Epimorphism of linear transformation

Find values of parameter t for which transformation is epimorphic: $\psi([x_1,x_2,x_3,x_4])=[x_1+x_2+x_3+2x_4,x_1+tx_2+x_3+3x_4,2x_1+x_2+tx_3+3x_4] $ When this transformation is epimorphic i.e. what ...
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1answer
68 views

Derivability of a function with an infinity of zeroes

Let $F$ be a normed vector space and $a\in F$. Is there a non zero function $f:\mathbb{R}\rightarrow F$, such that $f'(a)=0$ and $f$ is $0$ an infinity of times in any neighborhood of $a$ ? If not, ...
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1answer
38 views

Faulty proof that $V=U_1 \oplus W$ and $V=U_2 \oplus W$ implies $U_1 = U_2$

The question is as follows: Prove or give a counterexample: if $\ U_1, U_2, W$ are subspaces of $V$ such that $V=U_1 \oplus W$ and $\ V = U_2 \oplus W$, then $\ U_1 = U_2$. I happily ...
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Show that there exists a Hermitian form of signature $(p,q)$.

Let $K = \mathbb{Q}(\sqrt{-2})$ with $V_K = K^n$ considered as a $K$-vector space. Suppose $p, q \in \mathbb{Z}_{>0}$ such that $p + q = n$. Show that for any such $p$ and $q$ there is a Hermitian ...
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2answers
28 views

Prove vectors create a basis

Let $V$ be a vector space and $U,W,Z$ be it's subspaces where $V=Z \oplus U=Z\oplus W$. We know that $\beta_1,...,\beta_k$ is a basis of $U$ and $\beta_i=\gamma_i+\delta_i$ where $\gamma_i \in Z$ and ...
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0answers
23 views

Give the following linear transformation find values of parameter

Find values of parameter t for which transformation is epimorphic: $\psi([x_1,x_2,x_3,x_4])=x_1+x_2+x_3+2x_4,x_1+tx_2+x_3+3x_4,2x_1+x_2+tx_3+3x_4 $ When this transformation is epimorphic i.e. what ...