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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Let $a_1, …,a_n , b_1,…b_n$ be $2n$ distinct elements of a field , then is the matrix $\Big(\dfrac1{a_i-b_j}\Big)_{ij}$ non-singular?

Let $a_1, ...,a_n , b_1,...b_n$ be $2n$ distinct elements of a field and define $$h_{ij}:=\dfrac1{a_i-b_j} , \forall i,j=1,2 ,\dots,n. $$ Is the $n \times n$ matrix $H:=(h_{ij})$ non-singular ?
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If some vectors in $\mathbb Q^n$ are linearly independent over $\mathbb Q$ , then are they also linearly independent over $\mathbb C$?

Let $\vec v_1 , ..., \vec v_k $ be vectors in $\mathbb Q^n$ linearly independent over $\mathbb Q$ , then is it true that $\sum_{i=1}^ka_i\vec v_i=0, a_i\in \mathbb C, \forall 1\leq i\leq k \implies ...
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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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Why is the × operator not defined for vector × vector?

For a vector space, the + operator maps two vectors to another vector while the × operator maps a scalar and a vector to another vector. To me, it seems strange that scalars are seen as separate to ...
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389 views

What is the meaning of superscript ⊥ for a vector space

This should be an easy question, if A is a matrix, then the nullspace of A is a vector (sub)space. Then, what is the meaning of superscript inverted T on a vector (sub)space? e.g. $(\mathrm {nullspace ...
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linear map $f:V \rightarrow V$, which is injective but not surjective

I am trying to find a linear map $f:V \rightarrow V$, which is injective but not surjective. I always thought that if the dimension of the domain and codomain are equal and the map is injective it ...
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114 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 ...
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796 views

Why orthogonal basis?

Lets take the $\mathbb{R}^3$ space as example. Any point in the $\mathbb{R}^3$ space can be represented by 3 linearly independent vectors that need not be orthogonal to each other. What is that ...
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Linear operators on $C^\infty[a,b]$

I do not know too much about linear operators so forgive me if this doesn't make much sense, but what would the space of linear operators on $C^\infty[a,b]$ be defined as? If we denote this space as ...
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What “is” a matrix in the context of a vector space?

I'm familiar with the definition of a vector space $V$ over a field $F$ I'm also comfortable with the notion that a matrix "represents" a linear map from one vector space $V$ to another vector space ...
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227 views

Is there a difference between abstract vector spaces and vector spaces?

I am following my Oxford syllabus and my next step is abstract vector spaces, in my linear algebra book I've found vector spaces. I've searched a little and made a superficial comparison between ...
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Find the equation of the plane passing through a point and a vector orthogonal

I have come across this question that I need a tip for. Find the equation (general form) of the plane passing through the point $P(3,1,6)$ that is orthogonal to the vector $v=(1,7,-2)$. I would ...
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258 views

A problem : subspaces of a vector space

Let $L$, $M$, and $N$ are subspaces of a vector space. Prove that following is not necessarily true. $L \cap (M + N) = (L \cap M) + (L \cap N) $ This problem is given in 'Finite dimensional vector ...
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Riemann Sphere and a Strange Vector Space Definition

I'm reading Fractals Everywhere by Michael Barnsley. On pp. 6-8 [1] he defines a linear space which, he says, "is also called a vector space." However, his definition of a linear space only requires ...
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178 views

Can we have something like an orthonormal basis for a finite dimensional normed space?

So I proved a certain theorem about finite dimensional inner-product spaces, but after completing the proof, I realized the only point where I used the idea of orthogonality was the construction of an ...
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Are there norms on $\Bbb{C}^m$ and $\Bbb{C}^n$ so that the norm $\Vert\cdot\Vert$ is a subordinate norm?

Denote $$\Vert A\Vert=\sum_{1\le j,k\le m}\vert A_{j,k}\vert$$ is cleary a norm over $M_{m,n}(\Bbb{C})$ but not a subordinate norm by taking the identity matrix $I$. So my question is: Can we make ...
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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 ...
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158 views

Intuition for “the existence of a basis for every vector space is equivalent to the Axiom of Choice”?

Is there a intuitive way to understand "the existence of a basis for every vector space is equivalent to the Axiom of Choice"?
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Projection into a subspace?

Let $S$ be a nonzero subspace with orthogonal basis $(v_1, \ldots, v_k)$. Then the projection of $u$ onto $S$ is given by: $$\operatorname{proj}_S u = \frac {v_1 \dot{} u}{\operatorname{norm} ...
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Is the zero vector in $\mathbb{R}^n$ by itself a subspace of $\mathbb{R}^n$?

W is a subspace of $\mathbb{R}^n$ iff The zero vector ∈ W. X + Y ∈ W for any X, Y ∈ W. aX ∈ W for any X ∈ W and a ∈ R. So, given W = { X : X = [x1...], x1 = 0, x2 = 0, ... xn = 0 } ∈ Rn The zero ...
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Find the necessary and sufficient conditions on $A$ such that $\|T(\vec{x})\|=|\det A|\cdot\|\vec{x}\|$ for all $\vec{x}$.

Consider the mapping $T:\mathbb{R}^n\mapsto\mathbb{R}^n$ defined by $T(\vec{x})=A\vec{x}$ where $A$ is a $n\times n$ matrix. Find the necessary and sufficient conditions on $A$ such that ...
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357 views

Adjoint of forgetful functor between category of vector spaces and category of abelian groups

I've just found out about the forgetful functor between the category of vector spaces and the category of abelian groups. It maps a vector space to it's additive abelian group. My question is, is ...
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151 views

Finding the inverse of a matrix using a series

I want to find the inverse of the matrix $A$ given by: $ \left( \begin{array}{cc} 1 & -\epsilon \\ \epsilon & 1 \\ \end{array} \right) $ where $|\epsilon|$ $< 1$ (although I do not ...
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Proof of equivalence of algebraic and geometric dot product? [duplicate]

Geometrically the dot product of two vectors gives the angle between them (or the cosine of the angle to be precise). Algebraically, the dot product is a sum of products of the vector components ...
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243 views

How to get the upper triangular form for any linear map?

Let $T:V \to V$ be linear. $V$ is a complex vector space of dimension $k$. Then there exists a basis so that the matrix generated by $T$ under that basis is upper triangular. The proof is by induction ...
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401 views

Probability that $n$ vectors drawn randomly from $\mathbb{R}^n$ are linearly independent

Let's take $n$ vectors in $\mathbb{R}^n$ at random. What is the probability that these vectors are linearly independent? (i.e. they form a basis of $\mathbb{R}^n$) (of course the problem is ...
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1k views

What matrices preserve the $L_1$ norm for positive, unit norm vectors?

It's easy to show that orthogonal/unitary matrices preserve the $L_2$ norm of a vector, but if I want a transformation that preserves the $L_1$ norm, what can I deduce about the matrices that do this? ...
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Vector dimension of a set of functions

Let $F$ be a field and $S$ an infinite set. Set $V=\{f:S \rightarrow F\}$ endowed with the vector space structure that results from the pointwise operations of $F$. It is easy to prove that $|S| \leq ...
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direct products and direct sums for matrices and for vector spaces

I was wondering what relations and similarities are between direct product for matrices and direct product for vector spaces? Or do they just unfortunately and somehow misleadingly happen to have the ...
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93 views

What is needed to make Euclidean spaces isomorphic as groups?

Consider the abelian groups $G_n=(\mathbb R^n,+)$ for $n\geq1$. Claim: For any $n$ and $m$ the groups $G_n$ and $G_m$ are isomorphic. This claim is true if one assumes the axiom of choice, and I ...
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Is it possible to define an inner product such that an arbitrary operator is self adjoint?

Given a vector space $V$ (possibly infinite dimensional) with inner product $(.,.)$. We say an operator $A$ is self adjoint if $(Af,g)=(f,Ag)$. The definition as stated require us to start with an ...
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Is this condition sufficient to determine the linear space is of finite dimension?

From the Banach theory we knew that: 1) A linear space(a vector space endowed with its vector topology) $X$ of finite dimesion $dimX=n$ has the following property: If ${\left\| \bullet \right\|_1}$ ...
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Is there a name for the set of bit combinations of bitstrings?

Let $A \subset \{0,1\}^n$ be a set of $n$-bit bit vectors. Let me call a bit vector $b = (b^{(1)}, b^{(2)}, \dotsc, b^{(n)}) \in \{0,1\}^n$ a "bit combination" of the vectors in $A$ if: $$\forall i ...
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Generator of End(V)

If V is a finite-dimensional vector space of dimension n and G⊂End(V) such that G generates End(V) meaning that any element of End(V) is expressible as a linear combination of products of a number of ...
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135 views

Do all vectors belong to a vector space?

If we were given the vector $(1,1,1)$, say, we know immediately that it belongs to the vector space $\mathbb{R}^3$ (and infinitely many others). But, if we take this vector of dolphins: or some ...
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118 views

Are coordinate projections continuous?

Okay I have been working under the assumption that this is "obvious" for a while now, but it started to bug me and now I'm fumbling to prove it. Suppose $X$ is a normed linear space (possibly ...
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Infinite direct product of fields.

Let $F$ be a field, and consider the infinite direct product$$F \times F \times F \times F \times \dots,$$i.e. $\prod_{i=0}^\infty F$, i.e. the direct product of a countable number of copies of $F$. ...
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103 views

Duality mappings on finite-dimensional spaces

I have a few questions regarding some concepts in a book "nonlinear partial differential equations by Roubicek" I am studying. The following is from the text. "Let $V$ be a separable, reflexive ...
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87 views

Can something like $\text{Hom}(V,K)$ be visualised?

I have no trouble visualising vector spaces like $\Bbb R^3$ and (e.g.) a subspace of dimension $2$, which would just be a plane through the origin of a $3$-D space, but I'm having trouble visualising ...
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How to understand rank-nullity / dimension theorem proof?

OK, I am working on proofs of the rank-nullity (otherwise in my class known as the dimension theorem). Here's a proof that my professor gave in the class. I want to be sure I understand the ...
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Prove that the isomorphism between vector spaces and their duals is not natural [duplicate]

In preparation for an introductory talk on category theory, I recently spent some time thinking about natural transformations. The first example, or maybe the second, that everyone gives to motivate ...
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The basis of a matrix representation

If I have the linear map $f:\Bbb{R}^n\rightarrow \Bbb{R}^m$ then we can write $f$ as like the following: $$f\left(\vec x\right)=A\vec x$$ Where $A$ is a matrix. I think $A$ is called the standard ...
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Determinant of exact sequence

Let $0 \to A \to B \to C \to 0$ be an exact sequence of vector spaces. I want to show that I have a canonical isomorphism $$\text{det}(B)= \text{det}(A) \otimes \text{det}(C).$$ Here, "det" refers ...
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Example of infinite field $(F,+,.)$ such that $(F^*, . )$ is a cyclic group ? [duplicate]

It is known that any infinite cyclic group can never be a vector space , from this we can derive that if $(F,+,.)$ is an infinite field then $(F,+)$ cannot be cyclic . I am asking , is there any ...
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120 views

“Vector spaces” over a skew-field are free?

Are modules over a skew field free? That is, if $F$ is a skewfield then can any module $M$ be written as $\underset{i \in I}{\bigoplus} F$ for some indexing set $I$?
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Representations of Direct Sum of Lie Algebras

I'm trying to prove the following. Let $\frak{g}$ and $\frak{h}$ be (semisimple) Lie algebras. Then every representation $d$ of $\frak{g}\oplus\frak{h}$ is the tensor product of representations $d^1$ ...
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dimension of the space of all symmetric matrices with trace $0$ and $a_{11}=0$,

I want to know the dimension of the space of all symmetric matrices with trace $0$ and $a_{11}=0$, I can show that the dimension of space of all symmetric matrices $S$ is $n(n+1)/2$, now I give a ...
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Sphere tangent to a plane

Find the equation for a sphere with center $(\alpha,\beta,\gamma)$ tangent to the plane $ax + by + cz = d$. The sphere is $(x-\alpha)^2 + (y-\beta)^2 +(z-\gamma)^2 = r^2$ and I understand that some ...
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Limit of the projection of a matrix when the projection is not continuous

Consider two real matrices: the $n\times n$ matrix $A$ the $n\times m$ matrix $B$ of rank $m$, with $m<n$. Let, for $a\in\mathbb{R}$, $$S_a=A-aI_n,$$ and denote by $P_a$ the orthogonal ...
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Span and Dimension: A subspace

If $A$ is finite set of linearly independent vectors then the dimension of the subspace spanned by $A$ is equal to the number of vectors in $A$. This is obviously true. Since $A$ is a finite set of ...