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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4answers
69 views

Checking subspace

Let $B$ be a fixed matrix in $\mathbb{R}^{n\times n} $ and $W=\{{A \in \mathbb{R}^{n\times n} :AB=BA}\}$ Then is $W$ a subspace of $\mathbb{R}^{n\times n}$ ? I have tried this so far: a) The zero ...
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2answers
60 views

Does half-affine imply affine?

Let $V$ and $W$ denote real vector spaces, and consider a function $f : V \rightarrow W.$ Bind the variables $x$ and $y$ to $V$. Call $f$: half-linear iff for all real $a,b \geq 0,$ we have ...
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2answers
60 views

How to determine whether it is vector space? [closed]

Does the set of all polynomials of degree exactly $5$, together with all the constant polynomials,determine a vector space?
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1answer
65 views

My first proof related to subspaces (vector spaces). Please comment.

What do you think about my first proof which deals with subspaces? Theorem An intersection of subspaces is a subspace. Preliminaries Corresponding to the notation in Wikipedia, symbols for vectors ...
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1answer
68 views

The nullity of a square matrix with linearly dependent rows is at least one.

The nullity of a square matrix with linearly dependent rows is at least one. True or False? Here is the answer my textbook gives: True; if the rows are linearly dependent, then the rank is at ...
1
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1answer
61 views

Determine whether this polynomial form a vector space

I'm doing a question asking "A set of all polynomials with degree exactly 5. Does it form a vector space?" I'm a bit confusing showing multiplication part. If say the polynomial is ax^5+b, when the ...
0
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1answer
111 views

True or False: Matrices with linearly independent row and column vectors are square.

True or False: Matrices with linearly independent row and column vectors are square. Here is the answer of my textbook: True; if the row vectors are linearly independent then ...
6
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1answer
123 views

what is vector $(\vec{a}\cdot \vec{b})\vec{c} + (\vec{b}\cdot \vec{c})\vec{a} - (\vec{c} \cdot \vec{a})\vec{b}$

Suppose we have three non orthogonal vectors in $R^3$ as $\vec{a}, \vec{b}, \vec{c}$. The vector of $(\vec{b}\cdot \vec{c})\vec{a} - (\vec{c} \cdot \vec{a})\vec{b}$ is in the plane spanned by ...
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1answer
40 views

When does a matrix fail to be positive definite?

I am wondering how to think about a matrix being "bigger" than another. If I have the inequality X - Omega Sigma^-1 > 0 where all matrices are quadratic and X = Z'Z with Z positive definite and Omega ...
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1answer
51 views

Help explain linear algebra/differential calculus theorem in simpler terms.

On a previous question, I got something related to linear algebra and linear algebra, but having no background in linear algebra and a little background in vector calculus(mainly from physics), I ...
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1answer
124 views

Problem related to dual space of infinite dimensional v.space $V$

Let $V$ be a $K$-infinite dimensional vector space, and let $\mathcal B$ be a basis of $V$. For each $v \in \mathcal B$, let $\phi_v \in V^*$ given by $\phi_v(v)=1$ and $\phi_v(w)=0$, for all $w \in ...
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0answers
33 views

proof about real sequences

Let $V$ be the space of real sequences {${x_{1},x_{2},...}$} so that $\sum_{k=1}^{\infty}x_{k}^{2}$ converges. Let $W\subset V$ be the set of rational sequences with a finite number of terms. ...
0
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1answer
37 views

If $u$ and $v$ are vectors in $3$-space, then $u\cdot v$ is a scalar

My understanding is that B is definitely true because of the below picture but I cannot understand A. Please would someone point me to the right direction! Thanks!
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1answer
41 views

If $\mathbb{C}[x,y]/I$ is finite dim $\mathbb{C}$-vsp, does it have a monomial basis? Related to Hilbert Scheme of points in the plane.

Background (You can skip ahead if you wish): I'm trying to read this article about the Hilbert Scheme of points in the plane, and I don't understand one of the claims. An ideal $I\subset ...
1
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0answers
97 views

What do double vertical lines mean?

I am reading a paper on computer graphic and having hard time to understand this formula: What is the double vertical lines means? Do they always go with power of 2? If I want to learn further ...
3
votes
1answer
56 views

Sums of special vectors

Let $v$ be a vector obtained by taking a sum of $k$ vectors the of the form $(0,0,\ldots,0, -n, *,*,\ldots,*)$, where $"*"$ stands for either $0$ or $1$, and the position of the $-n$ entry can vary ...
6
votes
2answers
381 views

Linear algebra - Memorising proper definitions of homomorphism types

I am reading a book about linear algebra. On the basis of this book, I worked out the terminology below. Problem: To me, it looks like Wikipedia defines homomorphism differently. Apart from that: Do ...
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0answers
27 views

How to treat null elements in element-wise vector multiplication

we need to multiply for statistical reasons two huge normalized (0-1) vectors via element-wise or coordinate-wise multiplication. These two vectors are of different dimension (they are word ...
1
vote
1answer
78 views

What kind of space is this: $\Bbb{R}^n\times\Bbb{S}_{++}^n$?

Let $\Bbb{R}^n$ be the Euclidean space of $n$-dimensional column vectors with real coefficients. Moreover, $\Bbb{S}_{++}^n$ be the space of symmetric positive definite $n\times n$ real matrices. We ...
2
votes
3answers
64 views

Vector spaces - Non-uniqueness of element with property of scalar-multiplicative identity element?

I am dabbling in vector spaces, thinking about the axioms on Wikipedia. Notably, $$1 \mathbf{v} = \mathbf{v},$$ i.e. identity element of scalar multiplication (IEOSM), attracted my attention. I am ...
0
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0answers
62 views

prove that $\sum_{k=1}^\infty|x_k y_k|$ converges

Let $V$ be the space of real sequences $x_k$ so that $\sum_{k=1}^\infty x_k^2$ converges. Let $\langle x,y\rangle=\sum_{k=1}^\infty x_k y_k$ Prove that $\sum_{k=1}^\infty |x_k y_k|$ converges My ...
4
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1answer
79 views

What is the relationship between the trace/norm of a quaternion and the definition in field theory?

I'm having some trouble figuring out the relationship between the trace/norm of a quaternion element and the definition of trace/norm in the extensions of vector spaces. According to my number theory ...
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2answers
1k views

How to proof equality of del dot a cross b

I am trying to prove that $\nabla \dot{}(A\times B) = B\dot{}(\nabla \times A) - A\dot{}(\nabla\times B)$ I tried expanding the RHS but the $x$ component of the vector I am getting is ...
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1answer
37 views

Maximising a sum with respect to the unit ball.

Suppose that I have a vector $\boldsymbol{v} \in \mathbb{R}^d$, for some dimension $d>1$, and suppose I want to consider the sum $$ \begin{align*} \left(\sum_{k=1}^{d}v_k\right)^2. \end{align*} $$ ...
6
votes
3answers
178 views

When is $V=U\oplus U^{\perp}$?

Let $V$ be a (infinite dimensional) vector space with inner product $(,)$ and $V$ may not be complete with the metric induced from the norm. Let $U$ be a subspace of $V$. What is the necessary and ...
0
votes
2answers
42 views

Same column space is equivalent to same row space?

If $A$ and $B$ are $n \times n$ matrices that have the same column space, then $A$ and $B$ have the same row space. Can one prove or disprove this? This is my continuation of Same row space is ...
0
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1answer
199 views

Same row space is equivalent to same column space?

If $A$ and $B$ are $n \times n$ matrices that have the same row space, then $A$ and $B$ have the same column space. This is false of course. I could just come up with examples though. Can one prove ...
0
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1answer
27 views

Let $V$ be the space of real sequences {${x_{1},x_{2},…}$} so that $\sum_{k=1}^\infty {x_{k}}^{2}$ converges. Prove that this space is not numerable

Let $V$ be the space of real sequences {${x_{1},x_{2},...}$} so that $\sum_{k=1}^\infty {x_{k}}^{2}$ converges. Prove that this space is not numerable: My attempt: I have already proved that this is ...
3
votes
1answer
63 views

Sum of Neighborhoods of Zero

When do two neighborhoods of zero over a topological vector space add up as: $$aN+bN=(a+b)N\quad a,b\geq 0$$ I could imagine something like balanced might suffice... The problem is that I'd like to ...
2
votes
2answers
103 views

Vector Spaces: canonical basis for the usual vector spaces

I'm looking for a listing of the canonical basis for the most common vector spaces. Some standard basis are not so obvious. For instance, the basis for vector space $\Bbb C^2$ is $\{ ...
0
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1answer
29 views

Prove the number of unordered pairs of linearly independent elements

Let $V$ be a vector space over $K$. Let $K={\mathbb{Z}}/{p\mathbb{Z}}$, and $\dim V=3$. We know that $V$ has $p^3$ elements. I need to show that the number of unordered pairs of linearly ...
0
votes
1answer
50 views

Calculating the divergence of a central vector field.

I am trying to calculate the divergence of a central electric field, namely the electric field due to a point charge and my book begins like this: http://imgur.com/bW9tPEZ However in the last line I ...
1
vote
1answer
117 views

Does one need the Hahn-Banach theorem to prove the mean value inequality for maps into a normed space?

Consider the following mean value theorem: If $f$ is a continuous mapping of $\,[a,b]$ into a normed linear space $X$, whose norm doesn't derive from an inner product, and $f$ is differentiable on ...
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3answers
164 views

High dimensional vector space references

Is there any good text book or review papers that introduce high dimensional vector spaces and its peculiarities as compared to generic/low-dimensional vector spaces? For example, high dimensional ...
0
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1answer
41 views

Vectors-Can anyone explain me the concept of sense in vectors?

Is it same as the direction? Then, why another term "sense"is used, instead of direction? Can anyone illustrate it?
5
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1answer
66 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
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2answers
337 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 " . ...
2
votes
1answer
84 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
138 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 ...
0
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1answer
71 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 ...
0
votes
2answers
127 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
395 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
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1answer
1k 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
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0answers
72 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
46 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 $\Bbb R^3$ ...
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0answers
93 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
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1answer
54 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
115 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
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1answer
50 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
51 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 ...