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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Cardinality of Mappings on a 2-Sphere

I am wondering about the number of mappings from a point on a sphere to a neighboring point, and a not so neighboring point. If I take a 2-sphere, and place it on some $x,y,z$-axis and fix those so ...
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171 views

$(\mathbb R^2,+)$ as vector space over $\mathbb R$

It is not hard to see that $(\mathbb R^2,+)$ with this product $ {r\cdot(x,y)=(rx,ry) } $ is vector space over field $\mathbb R$. I'm looking for another product that $(\mathbb R^2,+)$ is ...
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252 views

Unit vector and directions

I am having problem understanding vectors. If a unit vector points in the direction of $z$ axis, then what coordinates would it have? The paper I read says $x$ and $y$ but if it is in the direction ...
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260 views

Prove Axiom $10$ (Vector Spaces) independent of the others [duplicate]

Possible Duplicate: Is it possible to construct a quasi-vectorial space without an identity element? In Apostol Multivariable Calculus, $1.5$ exercise $30 b$, he asks the reader to prove ...
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1answer
24 views

Requiring some light on the structure of the symplectic space and a given integer

Let $V$ be a vector space of dimension $2n$ and $f$ be a non-degenerate skew-symmetric bilinear form on $V$. $V'$ is a subspace of $V$, and the restriction of $f$ on $V'$ is of rank $2k$ for an ...
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39 views

Recommend a space to analyze the bearing of the vector between any two points

In Euclidean space, given any two points, the vector connecting them can be characterized by length (distance) and direction (bearing). Now I am only interested in the bearing part. And I found it is ...
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Fast approximate construction of orthogonal system

Assuming I have $d+1\in\mathbb{R}^d$ points that are not unfortunately chosen (in which case I can just resample, correct me if I $d$ points are enough), then these should span the whole space. What ...
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0answers
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Is the basis vector of a rotated vector in $E^3$ transformed differently than the components of the vector?

Do the basis vectors of a rotated vector in $E^3$ transform differently than the components of the vector? I've recently come across a blog where someone rotated the i,j,k basis vector using the ...
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53 views

Small perturbations

Background: Let $x_1,\ldots,x_n$ be the variables satisfying the equations of motion $\ddot{x_i}=f_i(x_1,\ldots,x_n)$ for $i=1,\ldots,n$ We introduce a small perturbation such that $x_i(t)=x_i^0 ...
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272 views

Linearly dependent vectors over finite fields

My problem is as follows: Assume you have a vector space of dimension $(d + 1)$, with values over $GF(q)$. Every vector in this vector space can be regarded as an element of the extension field ...
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1answer
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Working with a parallelepiped to find volume, area, angles

I am presented with the following problem: I have a parallelepiped with adjacent edges $$\vec{u} = [3,2,1]\\ \vec{v} = [2,3,1]\\ \vec{w} = [1,3,3]$$ a) Find volume b) find area of face ...
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207 views

How to prove linear independence using linear functionals in dual space?

I'm reading deBoor's (wonderful) book "A practical guide to splines", revised edition. I'm doing some of the exercises at the end of each chapter just to fix the main ideas before going ahead... ...
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86 views

what is mathematical difference between an hermitian operator $\hat A$ and a vector $\vec A$?

what is mathematical difference/relation between an hermitian operator $\hat A$ and a vector $\vec A$?
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92 views

Verify $X$ is a direct sum of $V$ and $W$

Let $$\begin{align*} X&=\mathcal{C}[0,1],\\ V&=\{v\in \mathcal{C}[0,1]\mid v(x)=v(-x)\},\\ W&=\{w\in\mathcal{C}[0,1]\mid w(x)=-w(-x)\}. \end{align*}$$ Is it possible to verify that $X$ is ...
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2k views

Subspaces of all real-valued continuous functions on $\mathbb{R}^1$

I'll go ahead and give you the problem first, and then explain my trouble with it. Which of the following subsets are subspaces of the vector space C(-$\infty$,$\infty$) defined as follows: Let V be ...
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1answer
132 views

Prove: If $L \leq X$, $L$ has finite dimension, $M\leq X$ Then $L+M$ is closed.

Prove: If $X$ is a locally convex space, $L \leq X$, $L$ has finite dimension, $M\leq X$ Then $L+M$ is closed. What I know: If $L$ is a finite dimensional subspace, then $L$ is closed.
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103 views

Need an example or contradiction.

If $A$ is a vector space then is it always true that $A+A=2A$. I know that it's not true but whats the use of saying no when i can't prove it. Thank you for your help.
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35 views

“Lock” a direction vector by preventing motion along a second vector

I have 2 unit vectors, o and v. o is the orientation of a cylinder and v is a direction I wish to move inside this cylinder. However, I want to allow v to only move perpendicular to the cylinder, so ...
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4answers
451 views

Prove that every real vector space has infinitely many vectors

I can't seem to wrap my brain around this one, so I figured someone here could point out the connection I'm not making. I've been asked to prove that every real vector space other than the trivial ...
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1answer
892 views

Shortest distance between point and line of intersection.

Find the shortest distance between the point $Q(11, 2, -1)$ and the line of intersection created by the planes x .$ \left(\begin{array}{cc} 1\\ -1\\ 3\\ \end{array}\right) = 0 ~$ and x ...
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1answer
26 views

Determine $\alpha$ for which this vector equation takes place.

Assume $ABCD$ is a parallelogram. $O$ is the intersection of the diagonals and $M$ an arbitrary point in the same plan. Determine $\alpha$ for which the following relation takes place: ...
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296 views

Exam Question: Find the dimension the subspace

I am putting an exam question of mine, which I hope complies with the overall policy of this forum. I have attempted it myself and I am checking effectively if my answer is right or not. EDIT: This ...
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Finding point distribution by eigen vectors

First of all I want to tell that my mathematics is poor, so I can’t use correct terms. Sorry for that. I have a point data set. This data represents some cylindrical objects surfaces (not exactly ...
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Determine whether the span of one set of vectors contains the span of another set of vectors

How can I determine whether the span of a set of vectors (such as $\mathrm{span}\{(3, 1), (4,1), (0,1)\}$ contains the span of another set of vector? EDIT: I realize that my original question was too ...
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2answers
99 views

Why doesn't this find the mid point?

I saw a simple question and decided to try an alternate method to see if I could get the same answer; however, it didn't work out how I had expected. Given $A(4, 4, 2)~$ and $~B(6, 1, 0)$, find ...
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947 views

How to prove that the union of two subspaces must be subsets of each other? [duplicate]

Possible Duplicate: Union of two vector subspaces not a subspace? $U,W\subseteq V$ are subspaces. Prove that in order for $U \cup W$ to be a subspace as well, either $U\subseteq W$ ...
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93 views

Introduction to Vectors

I am trying to write a hook for vectors on a linear algebra course. Does anyone have an opening hook for a section on vectors that will have a real impact on students?
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801 views

Prove $\mathbb{Z}$ is not a vector space over a field

This is an exercise from Chapter 3 of Golan's linear algebra book. Problem: Show $\mathbb{Z}$ is not a vector space over a field. Solution attempt: Suppose there is a such a field and proceed by ...
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262 views

how to prove fact about linear combinations of basis vectors using relationship between matrix columns?

How do we prove that if $\{v_1,...,v_m\}$ and $\{w_1,...,w_m\}$ are bases for a real vector space, then there are at most $m$ real numbers $\lambda$ such that $v_1+\lambda w_1,...,v_m+\lambda w_m$ are ...
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263 views

Finding the dimension of vector space V

I am trying to find the dimension of the vector space ($P_2$ means polynomials of degree at most $2$) $$V = \{p(x) \in P_2 \mid xp'(x) = p(x)\}.$$ However, I don't even know how to start... ...
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428 views

Vector space generated by the tensor products of pauli matrices

Let $\sigma_0,\sigma_x,\sigma_y,\sigma_z$ stand for the $2\times 2$ identity matrix and the well known pauli matrices: \begin{equation} ...
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1answer
29 views

For which points ($P$) on the $x$-axis, $\angle APB= 90^\circ $?

Let $A =(-2, 3, -2)$ and $B =(-6, -1, 1)$. For which points ($P$) on the $x$-axis, $\angle APB= 90^\circ $? I figured, since $P$ is supposed to be on the $x$-axis, the $y$ and $z$ coordinates ...
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1answer
104 views

Showing that $X=U\oplus W$

Let $X$ be a vector space, $V$ and $W$ are subspaces of $X$. Then I would like somebody to help on showing that $X$ is a direct sum of $V$ and $W$ if and only if $x\in X$ has a unique representation ...
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1answer
232 views

Does a basis span even linear systems?

I understood basis as a set of vector $v_{1},v_{2},...,v_{n}$ as the set whose linear combination will span the entire vector space say $ \mathbb R^{n}$ which makes perfect sense in intuitive terms. ...
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1answer
220 views

How is it possible for the inverse function of a linear-continuous-bijective function to be not continuous?

If $E$ and $F$ are two normed vector spaces, $f:E\rightarrow F$ is a linear-continuous-bijective function. Then naturally I would think that $f^{-1}$ is also linear-continuous-bijective. But the ...
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1answer
104 views

Is there a linear function that is *not* continuous between two normed vector space?

The textbook says that this function has to be continuous at least in the origin for it to be continuous everywhere. But how is it possible that a function is already linear but somehow not ...
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368 views

Vectors - Find the vector $\vec{c}$ which is orthogonal to $\vec{a}$ and $\vec{b}$ and whose first component is 1

Find the vector $\vec{c}$ which is orthogonal to $\vec{a}$ and $\vec{b}$ and whose first component is 1 $\vec{a}$ = (0 / 4 / -2), start point: $P$ (2 / -3 / 5), end point: $A$ (2 / 1 / 3) ...
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1answer
118 views

Inner product of two vectors?

When calculating the inner product$^1$ of two complex vectors $u$ and $v$, why is the complex conjugate of $v$ used? Why not just compute the inner product as with real vectors? 1:Where the inner ...
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3answers
1k views

Angle between two vectors?

I have been taught that the angle between two vectors is supposed to be their inner product. However, the book I'm reading states: Recall that the angle between two vectors $u = ...
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2answers
224 views

Vectors - Define a vector of length 1 orthogonal to $\vec{v} = (-4 \qquad 3)^t$

Define a vector of length 1 orthogonal to $\vec{v} = (-4 \qquad 3)^t$ I'm looking for the solution in terms of $\vec{a} = \binom{x}{y}$. How do I go about it? I'm familiar with addition, ...
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2answers
3k views

Proving if a given set is a vector space

I am working from Erwin Kreyszig's book, where he mentions of a question. Is the given set of vectors a vector space. If yes, determine the dimension and find a basis $(v_{1},v_{2},\cdots,v_{n} )$ ...
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3answers
167 views

Finding the space between two vectors?

Could someone explain why the formula: $$\theta = \cos^ {-1}(a \cdot b)$$ provides the angle between vector $a$ and vector $b$? All the online resources seem to explain how to find the angle, but not ...
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Interior of a Subspace

There is a conjecture: "The only subspace of a normed vector space $V$ that has a non-empty interior, is $V$ itself." (here, the topology is the obvious set of all open sets generated by the metric ...
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Boundedness of Surfaces in $\mathbb R^3$

GIven an equation such as $ax^2+by^2+cz^2+dxy+exz+fyz=g$ where $a,b,c,d,e,f,g\in \mathbb R$, How can we tell if the surface described is a bounded one without explicitly plotting a graph?
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1answer
59 views

Intuitive interpretation of this map

Could someone please explain to me what the map, $g$, is doing intuitively? $f:V\to V$ is a complex linear map and $f^n=\operatorname{id}$, for some $n>1$ $A$ is a subspace of $V$ and ...
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204 views

Vector Analysis & Linear Algebra

I'm given a positive number, a unit vector $u \in \mathbb{R} ^n $ and a sequence of vectors $ \{ b_k \} _{ k \geq 1} $ such that $|b_k - ku| \leq d $ for every $ k=1,2,...$. This obviously implies $ ...
2
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1answer
1k views

Equation for non-orthogonal projection of a point onto two vectors representing the isometric axis?

Suppose I have two vectors that are not orthogonal (let's say, an isometric grid) representing the new axis. Suppose I want to project a point onto these two vectors, how would I do it? Dot product ...
2
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1answer
416 views

A basic question about sum of two subspaces

Let $A$ and $B$ be a two subspaces of a vector space $V$ such that sum $A + B$ is not the whole of $V$. Then, can we say that there must exist a non zero vector $w$, orthogonal to every vector of ...
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1answer
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If $a=b+ c$ and $E$ is the basis of $a$, Will $E$ be also basis for $b$ and $c$?

Suppose $a$ lies in the span of a set of independent vectors $E$. Now, if $a=b+c$, is it also the case that $b$ and $c$ lie in the span o the same set of vectors $E$? if the question is obscure, ...
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160 views

Clarification: Viewing $\mathbb{R}^n$ as a probabilistic state space

In this MathOverflow post on visualizing high-dimensional spaces, Terry Tao states that "the fact that most of the mass of a unit ball in high dimensions lurks near the boundary of the ball can be ...