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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94 views

Show that $T\to T^*$ is an isomorphism (Where T is a linear transform)

I think I solved it, but I used a dirty trick, I'd like someone to review it, that would be great. Let $X,Y$ be linear spaces over field $F$. and $T:X \to Y$ a linear transformation.For each $T$ we ...
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1answer
102 views

Distance from Vector to the Linear Span

Let $V$ be the space of real polynomials of degree $\leq n$. a) Check the setting $(f(x),\,g(x))=\int_{0}^{1}f(x)g(x)\,dx$ turns $V$ to a Euclidean space. b) If $n=1$, find the distance from ...
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2answers
112 views

Vector space of real numbers over the rational numbers

What is the easiest way to show that $\mathbb R$ is not finitely generated over $\mathbb Q$ ?
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1answer
146 views

When is a vector space (over field $K$) also a ring (with subring $K$)?

(Apologies in advance for the very naive question. I'm just learning about all this. Also, for the sake of expedience, below I use the word "ring" when it would more correct for me to use ...
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1answer
105 views

Show that every finite-dimensional topological vector subspace is closed.

Let $X$ be a normed topological vector space. Show the following: (i) If $0\neq v \in X$, then $\{\alpha v:\alpha\in \mathbb{R}\}$ is closed. (ii) If $Y$ is a closed vector subspace of $X$ and $w\in ...
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1answer
43 views

Missing component of a 4D vector

I need to calculate the missing component of a 4D vector, when I know that one of the dimensions is always positive and less than or equal to the magnitude. In other words, I have four variables x, ...
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1answer
336 views

4D-vector calculations

For a 4D vector, how can I calculate any component as a function of the three other components and a magnitude and vice versa? I want x = f(y, z, i, m) where m is the magnitude of the vector. Will ...
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1answer
98 views

Is there a metric space and meanwhile a linear space such that vector addition discontinuous but scalar multiplication operation continuous?

Some special problems about topological groups or topological linear space theory. Recently I have done some study in some respects about topological group, topological linear spaces. And I found it's ...
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1answer
100 views

Proving Density of Subset of Hilbert Space

Suppose we have a subspace, $M$, of Hilbert space $H$. Prove the first statement implies the second statement: 1) If $<f,g> = 0$ for any $g\in M$, then $f=0$ in $H$. 2) $M$ is dense in $H$. I ...
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3answers
37 views

Question regarding ideals and vector spaces

this is my first time I am posting on this forum. My question is regarding a sentence I read on page 27 of "Algebraic Number Fields" by "Gerald J. Jansuz". The set-up is as follows: Let $R \subset ...
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1answer
599 views

All Invariant Subspaces of a Linear Transformation

I got this problem: Let $T:\mathbb{R}^3 \to \mathbb{R}^3$ be a linear transformation such that all it's eigenvalues are 1, 2 and 3 and the corresponding eigenvectors are $v_1, v_2$ and $v_3$ ...
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2answers
43 views

Product of invertible matrix and basis

I am confused about this problem. Let $S$ be a subspace of $\mathbb{R}^k$ of dimension $m\leq k$ and $\{b_1,...,b_m\}$ is a basis of $S$. Now, given an invertible matrix $A$. I have a feeling the set ...
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1answer
152 views

Question from Halmos' Finite-Dimensional Vector Spaces

This is part of question 6 in the set of exercises following section 17 in Halmos' Finite-Dimensional Vector Spaces. Suppose that $m < n$ and that $y_1,\dots,y_m$ are linear functionals on an ...
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1answer
123 views

Linear Transformation defined by a Matrix and Invariant Subspaces

I got stuck solving this problem: Let $T:\mathbb{R}^3\to \mathbb{R}^3$ be the linear transformation defined by the matrix A in the standard basis of $\mathbb{R}^3$, $E=\{e_1,e_2,e_3\}$ ...
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2answers
62 views

Direct sum of subspaces of the three dimensional space

$\newcommand{\span}[0]{\mathrm{span}}$I got stuck showing the following problem: If $\mathbb{R}^3 = W\oplus U$ where $W=\span\{e_1\}$ then $U = \span\{e_2,e_3\}$ I tried this way: Since ...
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1answer
165 views

Is there any non-translation invariant but homogeneous metric linear space?

A metric linear space is a metric space and vector space, and linear operation is continuous regarding to the metric. I know that a homogeneous, translation invariant metric $d$ can be used to define ...
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3answers
107 views

determinant of the linear transformation $T(X) =\frac{1}{2} (AX+XA)$

Let $V$ vector space of all matrices $3\times3$, and let $A$ be the diagonal matrix : $$ \begin{pmatrix} 1 & 0 & 0\\ 0 & 2& 0 \\ 0 & 0& 1\end{pmatrix} $$ Compute thee ...
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1answer
52 views

Pairwise Maximum Metric

Well , I had question on vector spaces and maximum metric . Lets us assume a set of vectors of $N$ dimension containing only integers , and let us make a set of vectors then we will calculate the ...
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1answer
403 views

If every subspace of a vector space V is invariant under a linear transformation T then T is a scalar transformation

I got the following problem Let $V$ be a vector space over field $\mathbb{F}$ and let $T:V \to V$ be a linear transformation such that every subspace of $V$ is invariant under $T$, Show that there ...
5
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2answers
78 views

Necessary condition for have same rank

Let $P,Q$ real $n\times n$ matrices such that $P^2=P$ , $Q^2=Q$ and $I-P-Q$ is an invertible matrix. Prove that $P$ and $Q$ have the same rank. Some help with this please , happy year and thanks.
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3answers
96 views

Matrix-free proof of $Z(GL_n(F)) = \{\lambda I:\lambda \in F^\times\}$?

How does one prove that $$Z(GL_n(F)) = \{\lambda I:\lambda \in F^\times\}$$ without resorting to matrices (and bases)? (BTW, $Z(GL_n(F))$ is the center of $GL_n(F)$, the general linear group of order ...
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1answer
114 views

Real Vector Space Scalar multiplication

If I have a real vector space, when are left and right scalar multiplication identical? I'm coming from the angle that the quaternions form a 2 dimensional vector space over $\mathbb{C}$ and yet left ...
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1answer
51 views

How to build a basis for a vector space E(n+1) from a set of points given in E(n) (a vector space of rank n).

I'm interested in how (and if) one can build a new dimension from a set of given dimensions. Specifically, if we are given a vector space E(n) of rank n, and a sample S of elements of E(n) (let us ...
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1answer
47 views

Linear transformations of the real polynomial space

Let n a natural fixed number and X the space of all real polynomials of degree at most n. I need to give a basis for X and say what of these following transformations are linear in X in X, this is ...
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3answers
128 views

Dual spaces, Find a natural isomorphism between V and $(V^{*})^{*}$

Let me just start by saying I'm very very new to this material. I have very little idea what's going on. I've red wikipedia and a few other sources but this is still very hard for me, so I would ...
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1answer
61 views

Subspace of Division Algebra

I'm working on understanding the following proof: https://dl.dropboxusercontent.com/u/17606191/proof.gif but I'm having some trouble understanding some of the author's terminology. We're asked to ...
3
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2answers
996 views

How to determine vector space?

I am taking a linear algebra course, and we are currently learning about vector spaces and subspaces. On the beginning of the chapter it is said that vector space must "comply" with all of the ten ...
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2answers
268 views

necessary condition for subspace of a vector space

Currently I'm reading linear algebra books of leon's and friedberg's. In friedberg's book, for being subspace, a subset of vector space should (1). contain zero vector (2). closed under scalar ...
3
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2answers
282 views

Easy visualizations of small countable ordinals

The ordinal number $\omega^2$ can be visualized as $\omega$-many copies of $\omega$. Likewise, the ordinal number $\omega^3$ can be visualized as $\omega^2$-many copies of $\omega$, arranged as ...
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2answers
197 views

The Hairy ball theorem and Möbius transformations

I just came across a chapter in Needham's Visual complex analysis; in particular, these diagrams: (p. 153 - these happen to be on the cover as well) They represent families of Möbius ...
3
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1answer
131 views

Why the space of skew-symmetric tensors $\Lambda^{n}V$ is a one dimensional if $dim(V)=n$

While reading Liviu Nicolaescu Lectures on the geometry of manifolds, I came accross the notion of "determinant line": Definition: Lev $V$ be an n-dimensional R-vector space. The one dimensional ...
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1answer
89 views

Are the 2nd order linear differential equations vector space?

Consider a homogeneous 2nd order linear differential equation $$a(x)y''(x) + b(x)y'(x) + c(x)y(x)=0,$$ where $a$, $b$, and $c$ are given functions of $x$. Let $V$ be the set of all real solutions ...
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1answer
240 views

Unit Vector Based on Angle with XY-YZ-XZ Planes

this may be a simple one but lets assume I have 3 angles (a,b,c) and I want to know what unit vector makes such angles with the XY-YZ-XZ planes. Another question is that I wanna know if a,b and c are ...
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1answer
1k views

Finding Rotation Axis and Angle to Align Two “Oriented Vectors”

In general, one can align a 3D vector $\vec A$ to another 3D vector $\vec B$ by rotating $\vec A$ around the axis $\| \vec A \times \vec B \|$ by the angle $\arccos{(\| \vec A \| \cdot \| \vec B ...
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0answers
45 views

Vector transforms

I have used ...
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1answer
2k views

Problem with line equation with double equal signs

My excersice is a lot bigger but I really need help with this line equation x-2=4y=z+1 Because it has two equal signs it's throwing me off a bit. Now my question is can I write the equation ...
2
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1answer
309 views

Prove that exist matrix P invertible then A=PB

Let two matrix $A=(a_{ij})_{m\times n}$ and $B=(b_{ij})_{m\times n}$ satisfy $\ker(A)=\ker(B)$ , $\: $($Ax=0\Leftrightarrow Bx=0$) Prove that exist matrix P invertible then A=PB. My tried: ...
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3answers
55 views

Let $S$ be a subset of $V$ . Identify which of the following statements is true:

Let $V$ be a vector space of dimension $d < \infty$, over $\mathbb{R}$. Let $U$ be a vector subspace of $V$ . Let $S$ be a subset of $V$. Identify which of the following statements is true: ...
2
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2answers
106 views

Square vs non-square tensors?

In mathematics, tensors are objects that operates on vector space. In physics or engineering, tensors usually operates on one vector space and its dual space: $V^{*} \times V^{*} \times V^{*} \times ...
37
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1answer
683 views

In categorical terms, why is there no canonical isomorphism from a finite dimensional vector space to its dual?

I've read in several places that one motivation for category theory was to be able to give precise meaning to statements like, "finite dimensional vector spaces are canonically isomorphic to their ...
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3answers
83 views

Determine the value of k such that the points A(4,-2,6), B(0,1,0), C(1,0,-5) and D(1,k, -2) lie on the same plane.

A(4,-2,6) B(0,1,0) C(1,0,-5) D(1,k, -2) if they lie on the same plane. How can i determine this? How do you know that the points lie on the same plane? Like do i check if they intersect? How ...
1
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0answers
112 views

Find linear independent vectors approximately

Given a set of vectors $a_1,…,a_n∈Z^m$ ($n,m > 10^5$ and $a_i∈Z^+ $ and about $n/2$ elements are zero). How can I approximate orthogonal vectors without using inner product? What vectors' features ...
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4answers
8k views

Determine if two straight lines given by parametric equations intersect

Does $[x,y,z] = [4,-3,2] + t[1,8,-3]$ intersect with $[x,y,z] = [1,0,3] + v[4,-5,-9] ?$ Attempt To find out if they intersect or not, should i find if the direction vector are scalar multiples? ...
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1answer
485 views

How to find $z$-intercept of vector equation

How do I find the $z$-intercept of the vector equation $\left<x,y,z\right> = (6, -2, -3) + t \left<3,-1,-2\right>$ I am so lost, do I set $x$ and $y$ equal to zero, and solve for $z$? I ...
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1answer
237 views

Question about Gram-Schmidt algorithm. Orthogonal diagonalization. Does GS conserve eigen-ness property

I have a question about the Gram-Schmidt process, and about the algorithm to find an orthogonal basis of eigenvectors (aka orthogonal diagonlization). let $T:V \to V$ be a diagonlizable linear ...
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2answers
1k views

Given the scalar equation, 8x + 9y = -45, write a vector equation?

scalar equation: 8x + 9y = -45 Attempt: I took the y-intercept and the x-intercept of the scalar equation and got (-5.625, 0) and (0,-5) By subtracting the points i got [5.625, -5] so my vector ...
1
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1answer
84 views

Notation for vector v in basis x, dimension y

What is common notation for the value of the $n$th dimension of vector $v$, given by basis $x$. Is it something like $$v_{x}^{y}$$ Thanks!
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3answers
407 views

Orthogonal complement in a finite field ${\mathbb Z}^{n}_{q}$

When $V=\mathbb{Z}^n_q$ is a vector space, where $\mathbb Z_q$ is the set of integers modulo prime $q>2$, are the following statements true? If $U ⊂ V$ is a $k$-dimensional ...
2
votes
2answers
58 views

Vector intersection

I have 2 vectors and their start points. i.e. $\vec p_1, \vec v_1$ and $\vec p_2, \vec v_2$ Now I want check if vectors intersect. I found this alghoritm. $\vec c = \vec p_2 - \vec p_1$ $\vec n_1 ...
0
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1answer
169 views

Let z1=x1+iy1 and z2=x2+iy2 be two complex numbers. The dot product of z1 and z2 is defined by <z1,z2>=x1x2+y1y2 [duplicate]

Let z1=x1+iy1 and z2=x2+iy2 be two complex numbers. The dot product of z1 and z2 is defined by z1,z2=x1x2+y1y2 For non zero z1 and z2 prove the following $$<z1,z2> =|z1||z2|\cosθ ...