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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What is the kernel of the tensor product of two maps?

Assume that $f_1\colon V_1\to W_1, f_2\colon V_2\to W_2$ are $k$-linear maps between $k$-vector spaces (over the same field $k$, but the dimension may be infinity). Then the tensor product ...
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Seeking an analytic proof of a vector identity

Show that for any vectors $\bf{u_1},\bf{u_2},\bf{v_1},\bf{v_2}\in\mathbb R^3$, we have $$(\bf{u_1}\times\bf{v_1})\cdot(\bf{u_2}\times\bf{v_2})= \left|\begin{matrix} \bf{u_1}\cdot\bf{u_2} & ...
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313 views

Every element of $U + V + W$ can be expressed uniquely in the form $u + v + w$

Suppose that $U$, $V$ and $W$ are subspaces of some given vector space. With the obvious definition of $U + V + W$, show that every element of $U + V + W$ can be expressed uniquely in the form $u + v ...
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623 views

Vector space of polynomials over $\mathbb{R}$ with degree $\leqslant n-1$

Let $P \in \mathbb{R}_{n-1}[X]$ be a polynomial of degree $n-1 \geqslant 0$. Let $\mathbb{R}_{n-1}[X]$ be the vector space of polynomials with degree $\leqslant n-1$ over $\mathbb{R}$. Show ...
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283 views

What are some examples of vector spaces that aren't graded?

From wikipedia: a vector space $V$ is graded if it decomposes into direct sum $ \oplus_{n \geq 0} V_n$ of vector spaces $V_n$. So as far as I understand things, any vector space with a countable ...
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What is the main difference between a vector space and a field?

In my opinion both are almost same. However there should be some differenes like any two elements can be multiplied in a field but it is not allowed in vector space as only scalar multiplication is ...
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Does there exist a unique definition of dot product in $\mathbb R^n$ such that the standard basis is orthonormal?

Does there exist a unique definition of scalar product in $\mathbb R^n$ so that the standard basis is orthonormal? I can't find a definition of dot product different from the usual definition. Is this ...
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526 views

If a field $F$ is such that $\left|F\right|>n-1$ why is $V$ a vector space over $F$ not equal to the union of proper subspaces of $V$

If $U_1$, $U_2,\ldots,U_n$ are proper subspaces of a vector space $V$ over a field $F$, and $|F|\gt n-1$, why is $V$ not equal to the union of the subspaces $U_1$, $U_2,\ldots,U_n$?
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Realification and Complexification of vector spaces

I am interested in a good comprehensive resource on realification and complexification of vector spaces over the reals or complexes (and the interplay of these structures on the 'same' space in ...
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plane determined by 2 vectors

i have 2 perpendicular vectors in space . How can i determine the plane determined by the 2 vectors? Regards, Alexandru Badescu
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Vector-space almost linear function

Find a function $f:V \rightarrow W$, where $V$ and $W$ are vector spaces (and V is defined on $\mathbb{K}$), such that $$f(x+y) = f(x) + f(y), \forall x,y \in V$$ but $$\exists a \in \mathbb{K}: ...
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Power-reduction formula

According to the Power-reduction formula, one can interchange between $\cos(x)^n$ and $\cos(nx)$ like the following: $$ \cos^n\theta = \frac{2}{2^n} \sum_{k=0}^{\frac{n-1}{2}} \binom{n}{k} ...
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6answers
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Given ${u, v, w}$ is a basis for $\mathbb{R}^3$, how can I show that $\{u + v + w, v + w, w\}$ is also a basis?

Given ${u, v, w}$ is a basis for $\mathbb{R}^3$, how can I show that $\{u + v + w, v + w, w\}$ is also a basis? I solved a similar problem in $\mathbb{R}^2$ (or at least think I did :p). ...
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Linear Algebra: determine whether the sets span the same subspace

So I am stuck on 51 here: 51. Determine whether the sets $S_1$ and $S_2$ span the same subspace of $\mathbb{R}^3$: $$\begin{align*} S_1 &= \Bigl\{ (1,2,-1),\ (0,1,1),\ (2,5,-1)\Bigr\}\\ ...
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693 views

Isomorphism of Vector spaces over $\mathbb{Q}$

From this post we see that $\mathbb{R}$ over $\mathbb{Q}$ is infinite dimensional. Similarly $\mathbb{C}$ over $\mathbb{Q}$ is also infinite dimensional, and I rememeber having solved a problem that ...
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51 views

Let $A_{j,k} = \langle x_j, x_k\rangle$. Show $A$ is invertible if and only if $x_1, \ldots, x_n$ are linearly independent.

Let $V$ be a vector space over $\mathbb C$ with inner product $\langle, \rangle$ and let $x_1, \ldots, x_n$ be vectors in $V$. Consider the $n \times n$-matrix $A$ with entries $A_{j,k} = \langle ...
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161 views

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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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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173 views

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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436 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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322 views

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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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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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 ...
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3answers
930 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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111 views

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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264 views

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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1answer
243 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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1answer
261 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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2answers
187 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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149 views

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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1answer
161 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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78 views

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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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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1answer
367 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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153 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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209 views

Cross products?

Say you have vectors $v$ and $w$. Let there cross product be denoted by $x$ so that: $$v \times w = x$$ According to Wikipedia: $$x_x = v_yw_z - v_zw_y$$ $$x_y = v_zw_x - v_xw_z$$ $$x_z = v_xw_y - ...
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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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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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251 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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427 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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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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4answers
129 views

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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1answer
858 views

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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410 views

Direct sum $\Rightarrow$ Direct Integral, Tensor product $\Rightarrow$?

Is there a way to define a tensor product over a measure space(=index set) with a continuous measure for Hilbert spaces? For the sum we have the notion of a direct integral, here.
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59 views

What is the implication that $\| \cdot \|_2$ and $\| \cdot \|_\infty$ are equivalent norms on $\mathbb{R^2}$

Given $\mathbb{X}$ = $\mathbb{R^2}$, consider $\| \cdot \|_2$ and $\| \cdot \|_\infty$ We can show that $\| x \|_\infty \leq \| x \|_2 \leq \sqrt2 \| x \|_\infty$ Hence $\| \cdot \|_2$ and $\| ...
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1answer
86 views

About $\mathbb{R}$ as a vector space over $\mathbb{Q}$

I want to understand better the structure of the vector space $\mathbb{R}$ over $\mathbb{Q}$. I know that it is an infinite dimensional vector space with a non countable Hamel basis, and it is cited ...
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1answer
151 views

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 ...