Questions tagged [vector-spaces]

For questions about vector spaces and their properties. More general questions about linear algebra belong under the [linear-algebra] tag. A vector space is a space which consists of elements called "vectors", which can be added and multiplied by scalars

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Method for decomposing an algebra into a direct sum of matrix algebras

I'm reading a paper which says that there is a standard way to decompose a (finite dimensional) algebra into a direct sum of matrix algebras by finding elements of the algebra which satisfy: $$e_{ij}...
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Connectedness of normed vector space

Is it possible to have a normed vector space $V$ over some field $F$ that is not connected? (I already know that if $F=\mathbb{R}$ or $F=\mathbb{C}$ then we can easily show that this is not possible. ...
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Example on basis change

I am reading the book Mathematics for Machine Learning by Marc Peter Deisenroth, A. Aldo Faisal, and Cheng Soon Ong. I have a problem understanding an example in this book: Example 2.23 (Basis change) ...
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Let $W$ be a $n$ dimensional vector space over $C$, and let $T \subseteq L(W)$

Prove the two following are equivalent: $W$ can be decomposed into direct sum of two non-trival T-invariant subspaces, $U$ and $V$. Jordan Form of $T$ consists of more than one Jordan block.
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1answer
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Proof that a Diffeomorphism Between Two Coordinate Systems Compatible with an Orientation on a Manifold has Positive Determinant

On pages 118-119 of Spivak's Calculus on Manifolds he introduces the idea of consistent orientations by stating It is often necessary to choose an orientation $\mu_x$ for each tangent space $M_x$ of ...
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Basis change qestion

Let $V$ be a vector space over a field $K$. Suppose we have two bases of $V$, $B=(e_1,...,e_n)$ and $B'(f_1,...,f_n)$. We can express $f_j$ in terms of the basis $B$: $$f_j=p_{1j}e_1+p_{2j}e_2+...+p_{...
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Homeomorphic to a vector space but not itself a vector space

In Nash & Sen p.162, they show that the space of all positive definite symmetric matrices $C$, while not a vector space itself, is homeomorphic to the space of all symmetric matrices $S$ (which is ...
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Related to Euclidean and unitary vector space

Let $V$ be a finite-dimensional Euclidean or unitary $K$-vector space. Show or refute the following statements: (i) $(f + g)^{*} = f^{*} + g^{*}$ for all $f, g ∈ \operatorname{End}(V)$ (ii) $(λf)^{*}$ ...
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1answer
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Minimizing the norm of the difference of two vectors.

Let $\mathcal{H}$ be a a vector space of finite dimension. Let $\mathcal{H_1}$ be a subspace of $\mathcal{H}$. Considering some vector $|\phi\rangle \in \mathcal{H}$ i need to show that there exists ...
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1answer
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Sum of four one-dimensional subspaces

I have this tricky problem: Consider four subspaces $E_1, E_2, E_3, E_4\subset\mathbb{R}^{4}$ such that: a) dim$(E_{k})=3$ for $k=1,2,3,4$. b) dim$(E_{i}\cap E_{j})=2$ if $i\neq j$, c) dim$(E_{i}\cap ...
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How do you find a basis for $\mathbb{C}$ as a vector space over $\mathbb{Q}(\zeta_n)$?

In the comments on a question I posted earlier it was recommended that I find a basis for $\mathbb{C}$ as a vector space over $\mathbb{Q}(\zeta_n)$. How should I go about this? What would be a good ...
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The intersection Grassmanian of $n-k$-planes and Grassmanian of $k$-planes

I am reading a paper that I don't understand some parts of it. Let $A$ be a $4\times 4$ matrix, which its char poly is irreducible. Then, the paper said we need to check "for or every pair of $A$-...
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1answer
25 views

Subspaces, condition on dimensions, orthogonal complement

Let $U$ be a finite dimensional vector space, endowed with an inner product $\langle\cdot,\cdot\rangle$, and let $V,W$ be some non-trivial subspaces of $U$. We know that $\dim W < \dim V$. Does it ...
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Let V be a vector space of dimension n and W a non trivial proper subspace of V. [closed]

Let V be a vector space of dimension n and W a non trivial proper subspace of V. A) what is the max dimension of V/W? Give an example of a quotient space V/W of that max dimension. B) what is the min ...
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Do exist infinitely many unitary transformations between two real vectors?

Are there infinitely many unitary transformations to satisfy the relation $U\left| b \right\rangle = \left| a \right\rangle $ (where $UU^{\dagger} = U^{\dagger}U=1$) between two arbitrary, normalized ...
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1answer
37 views

Cardinality of the sum of two sets

the question ist how to prove the following: $|X+Y|\ge |X|+|Y|-1$, where $X$ und $Y$ are two finite subsets in some vector space. I know the proof for the one-dimensional case but it is based on the ...
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Transformation matrix for a basis

I have the follow question here. Let $B=\{u_1,u_2,u_3\}$ with $u_1=1$, $u_2=x$, $u_3=x^2$ denote a basis for the space of polynomials of the second order polynomials $P^2$. Let $T(a_0+a_1x+a_2x^2)=...
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Generalizing dot product to arbitrary order

For a vector $\mathbf{r}$, if we define $\mathbf{r}^1 \equiv \mathbf{r}$ and $\mathbf{r}^2 \equiv \mathbf{r} \cdot \mathbf{r}$, we find that $\mathbf{r}^1$ is a vector and $\mathbf{r}^2$ is a ...
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What is the term for this operation on vectors?

Let's say I have a vector of integers such as $$ \langle1,23,3\rangle \\ \langle41,5,16\rangle \\ \langle3,5,7\rangle \\ \langle10,13,31\rangle $$ and I wish to create a new vector whose entries are ...
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Properties of Bilinear form

Let $K$ be a field, let $V$ be an $n$-dimensional $K$-vector space, and let $f: V \times V \to K$ be a bilinear form. The bilinear form $f$ is called non-degenerate if the mapping $$V \to \...
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2answers
36 views

Name of a vector of 1s?

Let's consider an N dimensional vector where each coordinate takes the value 1. For example, for $N=5$ we have: $(1,1,1,1,1)$. Does this type of vector have a name (unary?). Are there any conventions ...
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How does $x=2$ form a 2-Dimensional Space?

Today when professor solving old exam questions one question like; Vectors satisfying x=2 equation form a -- dimensional space in R^3. And he said blank will be 2-...
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1answer
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In Linear Algebra, does a vector equal its transpose

In this post: why do people say “x dimensional vector" a question regarding vector dimension was framed. The response was more or less: vectors have dimension equal to space of which the vector ...
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Help with finding a specific example of a Vectorspace

Im trying to find a Vectorspace $V$ that fulfills following requirements (if existent): $\|x\|_1 > \|y\|_1$ $\|x\|_2 < \|y\|_2$ such that $x,y$ are elements of $V$
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75 views

Ring, in which $1+1=0$ holds

First question comes from famous university's first examination of linear algebra. First question: Let $R$ be a finite ring, in which $1+1=0$ holds. Then, I would like to prove the number of elements ...
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1answer
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$||u||_2= \sup_{x \in K} |u(x)|$ is not a norm when $K$ is closed subset of $\mathbb R^k$ and $u(x)$ is a continuous function on $K$.

$K \subset \mathbb R^n$ and $K$ is closed. $C(K)$ is the set of all continuous function form $K$ to $\mathbb R$. Now for every $u(x) \in C(K)$ , $||u||_2= \sup_{x \in K} |u(x)|$. $||u||_2$ is ...
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Spotting a mistake in proving if $U_1 + W = U_2 +W$ then $U_1 = U_2$, where $W, U_1, U_2$ are subspaces.

I'm working through Axler's Linear Algebra Done Right, and despite being able to find a counter example, I cannot understand why my proof is incorrect for the following problem. Claim Let $U_1$, $U_2$,...
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$M^n/M^{n+1}$ is vector space over a residue field $k=R/M$

Let $(R,M)$ be a local ring. I heard that $M^n/M^{n+1}$ is vector space over a residue field $k=R/M$. I would like to confirm this is true. Firstly, Could you tell me how we define map $k×M^n/M^{n+1}→...
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Order of an element in a vector space over field of characteristic $p$

Let $p$ be a prime. Let $G$ be a vector space over field of characteristic $p$. Then, every element of $G$ has order of $p$ or $0$? I think this seems to be true, could you prove this? Thank you in ...
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1answer
29 views

Show case when minimal polynomial coincides with its characteristic polynomial

As is introduced in the title, I'm stuck on the following problem: Considering a linear endomorphism $φ$ of an $n$-dimensional vector space $V$ having $n$ pairwise distinct eigenvalues, I would like ...
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1answer
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Finding basis and explain why the basis have found is a basis

Let u = (1, 2, 3, 4) and v = (4, 3, 2, 1) be two vectors in $R^4$. These vectors define the subset of $R^4$ $V = \{x \in R^4 | u \bullet x = 0$ and $v \bullet x = 0\}$ Here $u \bullet x$ denotes ...
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1answer
36 views

Why is the following set open in the dual space?

Let $E$ be a normed vector space over $\mathbb{C}$, and consider $E^{\ast}$ the space of continuous linear functionals on $E$ with the topology induced by the norm $\|f\| = \sup_{\|x\| \leq 1 } |f(x)|$...
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Basis of Vector Space with Non-Standard Definition of Addition and Multiplication

Consider the vector space $V = \mathbb R^+ \times \mathbb R^+ \times\mathbb R^+$, where $\mathbb R^+$ is the set of all positive real numbers with addition and scalar multiplication defined by $(x_1, ...
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Determine whether the given set S is a subspace of the vector space V

(a) V=R^n, and S is the set of solutions to the homogeneous linear system Ax=0 where A is a fixed m×n matrix. (b) V=P^4, and S is the subset of P subscript 4 consisting of all polynomials of the form ...
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Literature on co- and contravariant formulation

I am currently working on the topic of co- and contravariant formulation in physics. Unfortunately, my literature uses the topic very superficially. Is there any mathematical book with definition, ...
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extrior product of a square matrix

I want to understand how one can find the exterior product of square matrices. I searched about it and I find a lot of complicated formulas and without any examples. On the other hand, I found this ...
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1answer
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Is there a geometrical meaning to inequalities about norms of orthogonal transformations?

Let $V$ be a normed vector space of a finite dimension $n$. We say that a linear map is orthogonal if $\| Av \| = \| v \|$ for every $v \in V$. The norm of a liner map is defined as $\|A\|=max_{v\in V,...
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1answer
41 views

Prove that $\mathbb{R^3}$ is not a vector space

Consider $\mathbb{R^3}$ with the usual addition $+$ of vectors, but with scalar multiplication $\otimes$ defined by: $k$ $\otimes$ $\begin{bmatrix} x \\ y \\ z \end{bmatrix}$ = $\begin{bmatrix} x \\ ...
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28 views

Annihilator of direct product

Let $E_1,E_2$ be two subspaces in $GF(2)^n$ such that $E_1\cap E_2=\{0\}$. If $E_1\times E_2=\{(x,y):x\in E_1,y\in E_2\}$, is it true that $$(E_1\times E_2)^{\perp}=E_1^{\perp}\times E_2^{\perp},$$ ...
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1answer
25 views

Dual Space of $\mathbb{R}^n$

I am trying to understand dual spaces. Suppose I have the Euclidean vector space $\mathbb{R}^n$ over the field $\mathbb{R}$. Elements of this space are column vectors $\boldsymbol{x}\in\mathbb{R}^{n \...
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Definition of expectation for non-real-valued random variables

I have checked many sources for the definition of expectation for non-real-valued random variables. I am interested on the conditions of the image space that guarantee the existence of an expectation ...
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2answers
40 views

How to show this ring is Noetherian?

I got the following exercise: Let $W$ be a finite-dimensional $\Bbb{R}$-vector space. Let $\Bbb{R}_W=\Bbb{R}\times W$. Define addition and multiplication by $(r,w)+(s,v)=(r+s,w+v)$, $(r,w)*(s,v)=(rs,...
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Basis Extension Theorem

I know that using basis extension theorem we can extend a set of LI vector to the basis of vector space, but how we actually do it? I have only theoretically used this theorem in proof, how to ...
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1answer
35 views

Prove or Disprove $\mathbb{R}^3$ is a vector subspace

I'm currently reviewing the topic of subspaces but I'm baffled in this problem. I don't understand the answer provided. May someone please expand on it.
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How to compute similarity score between two vectors of same size [N*d] and compare the result with a numeric value? [closed]

I have two sentences embeddings of size [N_words*dim]. I want to compute similarity between these embeddings. I also have a label of similarity score (decimal). I want to compare the predicted ...
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1answer
25 views

Subspace generated by three bivectors

I have the following exercise that I have not been able to solve: Consider three simple bivectors $B_i=\alpha_i\wedge\beta_i, i=1,2,3$ in $\mathbb{R}^{4}$. The bivectors $B_i$ are linearly independent ...
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Nonestraight segments in a vector space

Is there a vectorial structure on $\mathbb{R}^2$ such that the segments $[x,y]$ with boundary points $x, y \in \mathbb{R}^2$ are not graphically straight? $[x,y]$ is defined as $\{\lambda x +(1-\...
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What is the conventional way to define a norm-like measure on a vector space in which logarithmic function has poles? [closed]

Is there a conventional way to define norm-like measure and metric on a vector space in which logarithmic function has non-logarithmic singularities (poles) on some element? If the logarithm has a ...
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26 views

Is it possible that a vector has undefined magnitude/norm? [closed]

Is it possible that the magnitude or norm or pseudo-norm function has a singularity or discontinuity at some element of a vector space?
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1answer
41 views

How can you compare equality of two vectors of infinite dimension?

E.g., say we have 2 vectors of infinite dimension: $$A=<1,0,1,1,...,1>$$$$B=<0,1,1,1,...,1>$$ These are qualitatively not the same vector, though it seems hard to definitively prove that. ...

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