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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Difference between isometry and euclidean motion

I know that they both preserve distance however what is the difference between them? I think isometry is part of Linear Transformations and euclidean motion is not
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Zero element in Lie algebra of a Lie group

On page 189 of John Lee’s Introduction to Smooth Manifolds it is stated that the set of all smooth left-invariant vector fields on a Lie group $G$ is a linear subspace of the space of all smooth ...
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Column space of a linear model

Suppose I have the following linear system of equation $$X = AC$$ with $X$ of dimensions $N \times T$, $A$ of dimension $N \times D$ and $C$ of dimension $D \times T$. Does it hold in general that $\...
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Understanding proof of "linear dependence of functions $f_1, f_2,...,f_n$ implies Wronskian of these functions is identically zero

The proof is shown in below picture. I am not able to understand the underlined part. How author concluded that, "linear dependence of $f_1,f_2,...,f_n$ implies that the linear system in ...
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Compute the distance d from y to the straight line through u and the origin.

Let y = $$\begin{bmatrix} 3 \\ 1 \end{bmatrix}$$ and u = $$\begin{bmatrix} 8 \\6 \end{bmatrix}$$. Compute the distance d from y to the straight line through u and the origin. d = $$\begin{vmatrix} y - ...
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Theorem 6, Section 2.3 of Hoffman and Kunze’s Linear Algebra

If $W_1$ and $W_2$ are finite-dimensional subspaces of a vector space $V$, then $W_1+W_2$ is finite-dimensional and $\mathrm{dim}(W_1)+ \mathrm{dim}(W_2)= \mathrm{dim}(W_1\cap W_2)+ \mathrm{dim}(W_1+...
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1 vote
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Can I use the mapping notation to describe this vector space?

So, I recently learned that the statement $v \in \mathbf F^S$, where $\mathbf F$ is a field and $S$ is a set, can be rewritten as $v : S \rightarrow \mathbf F$. Does this imply that $ x\in \mathbf F^...
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Surjectivity and injectivity of tensor products

Let $\phi_j:V_j\rightarrow W_j$ be linear maps between $\mathbb{R}$-vector spaces for $j=1,...,k$ and define $\psi:=\phi_1 \otimes ... \otimes \phi_k : V_1 \otimes ... \otimes V_k \rightarrow W_1\...
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Proving that an intersection of affine spaces contains only a single vector

Let $V_1$ and $V_2$ be subspaces of $V$, such that $V=V_1 \oplus V_2$. Prove that the intersection of affine spaces $V_1 + a$ and $V_2 + b$ contains a single vector. I understand the intuition behind ...
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Showing that if $V$ is a vector space over $F$ such that $|V|>2$, then $V$ has more than one basis [closed]

I am self studying Advanced Linear Algebra where I came across the following exercise If $V$ is a vector space over $F$ for which $|V|>2$, then $V$ has more than one basis. Here, $|V|$ is the ...
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The zero function in a subspace [closed]

The solution for this question says, "The given subset does not contain the zero function" What do they mean by the zero function? I understand that any subspace must house the zero element. ...
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Is it true that every associative algebra is a subalgebra of a unitary associative algebra?

I know from this proposistion we can get "there is always a faithful representation of any associative algebra", but I have difficulty to prove it.
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∂/∂𝑥 and ∂/∂𝑦, are they elements of a Ring or a Vector Space??

As we know, in complex analysis we can define $${\partial _z} = {1 \over 2} ({\partial _x} + {1 \over i}{\partial _y}) \ \text{and}\ {\partial _\overline{z}} = {1 \over 2} ({\partial _x} - {1 \over i}{...
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Alternative Solution to Theorem 5 Corollary 2, Section 2.3 of Hoffman’s Linear Algebra

In this video lecture, time stamp 19:20 - 24:30. Professor show proof of following claim: (Extend to basis) Every linearly independent list of vectors in a finite dimensional vector space can be ...
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Size of linearly independent set bounded by the size of a spanning set

How would I go about proving this theorem? Let $V$ be a vector space over $K$, $\{u_1, ..., u_m\} \subseteq V$ be linearly independent & let $\{v_1, ..., v_n\}$ span $V$. Then $m \leq N$, (i.e ...
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Submodules of a complex vector space and invariants of an endomorphism

Let $V=\mathbb{C^n}$ be a finite dimensional vector space and $f\in End(V)$. Consider the $\mathbb{C[x]}$-module $\mathbb{C_f^n}$ which is obtained by giving $V$ the left-module structure as $p.v=p(f)(...
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What conditions must we put on a group in order for it to be a vector space over some field?

The axioms specifying the addition operation of a vector space are precisely those defining an abelian group, so of course it needs to be abelian. That's not sufficient though, as per this quora post.....
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How every subspace is the solution space of a homogeneous system of linear equations?

There is this line in book "Mathematics for Machine Learning": " Every subspace $U ⊆ (R^n , +, ·)$ is the solution space of a homogeneous system of linear equations Ax = 0 for x ∈ $R^n$ ...
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Theorem 5, Section 2.3 of Hoffman’s Linear Algebra

If $W$ is a subspace of a finite dimensional vector space $V$, every linearly independent subset of $W$ is finite and is part of a (finite) basis for $W$. Rephrasing Theorem to my taste: If $W\leq V$...
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2 votes
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Can vector spaces occupy a large cardinal amount of dimensions?

So far I have found that a vector space of uncountably infinite dimensions is mathematically valid, but what about vector spaces that can occupy, say, an inaccessible or Mahlo cardinal amount of ...
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Properties of homomorphism [closed]

Given f is an homomorphism from U to V, where U and V are two vector spaces, how can I prove the following propertis: If W is a third vector space, and g is an homomorphism from V to W, then is g ◦ f ...
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Finding a Curve in a Vector Field that has 0 Flow

I am going through Vector Calculus for the first time and I had a thought which I assume has a neat solution, but I could not find a good answer for it: Imagine I have given 2D vector field, I want to ...
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Relation between subspaces and polynomials

Let $E = \mathbb{R}_{2022}[x]$. If F and G are in E, and $r + s \leq 2020$ where r and s are natural numbers and $$F=\{P(x) \in E: (x-2)^r | P(x)\}$$ and $$G=\{P(x) \in E: (x-22)^s | P(x)\}$$ find: a)...
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How do you call it when you stack vectors on top of each other?

How do you call it when you stack two vectors, let's say $u=\pmatrix{u_1\\u_2}$, $v=\pmatrix{v_1\\v_2}$, on top of each other such that you get $$u\oplus v=\pmatrix{u_1\\u_2\\v_1\\v_2}?$$ I found this ...
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Theorem 4, Section 2.3 of Hoffman’s Linear Algebra

Let $V$ be a vector space which is spanned by a finite set of vectors $\beta_1,…,\beta_m$. Then any independent set of vectors in $V$ is finite and contains no more than $m$ elements. Question: (1) ...
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Geometric interpretation of Grothendieck inequality

The (real) Grothendieck inequality of order $d$ implies that for every $n\times n$ real matrix $M$ satisfying $$ \left \lvert \sum_{i,j=1}^n M_{i,j} s_i t_j \right \rvert \leq 1 \;, $$ with $s_i,t_j \...
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2 answers
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Basis for $\mathbb R^{n\times m}$ using $n$ linearly independent $n$-vectors

Exhibit a basis set for $\mathbb R ^{n\times m}$ for $n\geq m$. Obviously $$\begin{pmatrix}1&0&...&0\\0&0&...&0\\\vdots&\vdots&\ddots&\vdots\\0&0&...&0\...
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Can two different set of independent vectors span same plane?

Can two different set of independent vectors span the same plane? let's say two independent basis vectors v1 and v2 span a plane P1. Can we find different set of independent vectors v3 and v4 that ...
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3 answers
47 views

Smooth convex set intersects any vector non-coplanar with supporting hyperplane?

Suppose I have a convex, compact, and $n$-dimensional set $K \subseteq \mathbb{R}^n$, and the origin $o$ is a smooth point on the boundary of $K$. Let $b_1, ..., b_{n-1}$ be a basis for the unique ...
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1 vote
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Exercise 6, Section 2.2 of Hoffman’s Linear Algebra

(a) Prove that the only subspaces of $\Bbb{R}$ are $\Bbb{R}$ and the zero subspace. (b) Prove that a subspace of $\Bbb{R}^2$ is $\Bbb{R}^2$, or the zero subspace, or consist of all scalar multiples of ...
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4 votes
3 answers
136 views

Combinatorial problem using vector space

Let me state the problem first. This is an exercise from discrete math. Let $k$, $n$ be positive integers with $1 \leq k \leq n$. Let $\mathcal{F} = \{A_1,A_2, \cdots, A_m\}$ be a set of subsets of $\{...
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Is it true that the image of the relative interior is the relative interior of the image?

Suppose $X$ is a convex subset of a topological vector space. $f$ is a linear function. Is it true that $ri(f(X))=f(ri(X))$, where ri means relative interior? If so, is there a reference from which I ...
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1 vote
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An extension of Hahn-Banach theorem to the case where an affine function is dominated by a convex one

I'm trying to prove below extension of Hahn-Banach theorem. Let $X$ be a vector space, $f: X \to \mathbb R \cup \{+\infty\}$ convex, $A$ an affine subspace of $X$, and $g : A \to \mathbb R$ affine ...
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2 votes
0 answers
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Let $f,g$ be affine and $f', g'$ their induced linear maps. If $f \le g$, then $f' = g'$

Two parallel lines on $\mathbb R^2$ have the same slope. I'm trying to generalize this result to higher dimension. Could you have a check on my attempt? Let $X$ be a vector space and $A$ an affine ...
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Dimension of the direct sum

The direct sum of two vector spaces is $\mathcal V \oplus \mathcal W =\{v+w:v\in V,w\in W\}$. Show that $\dim(\mathcal V\oplus \mathcal W)=\dim(\mathcal V)+\dim(\mathcal W)-\dim(\mathcal V\cap \...
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0 answers
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The relationship between affine and linear maps

I'm reading a version of Hahn-Banach theorem for convex function. This version involves affine map. Could you confirm if my below understanding is correct? Let $X$ be a vector space and $A$ its ...
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Is my proof for any finite-dimensional vector space being isomorphic to its double dual correct?

Learning about dual vector spaces I began thinking about how the product of a row vector and a column vector acts like a dot product and that furthermore, the row vector is almost acting like a ...
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Exercise 9, Section 2.2 of Hoffman’s Linear Algebra

Let $W_1$ and $W_2$ be subspaces of a vector space $V$ such that $W_1+W_2=V$ and $W_1\cap W_2=\{0\}$. Prove that for each vector $\alpha$ in $V$ there are unique vectors $\alpha_1$ in $W_1$ and $\...
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1 vote
0 answers
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Inverting vector basis and how it affects vectors defined in standard basis

We consider here a vector basis $B_R$ of $\mathbb{R}^2$ that is derived from the standard basis $$ E=\left\{ \left( \begin{array}{c} 1\\ 0\\ \end{array} \right) ,\;\left( \begin{array}{c} 0\\ 1\\...
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0 answers
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Separation of a convex and a concave functions

I'm trying to prove below separation theorem, i.e., Theorem: Let $X$ be a vector space, $f:X \to \mathbb R \cup \{+\infty\}$ convex, and $g:X \to \mathbb R \cup \{-\infty\}$ concave. Assume that $$g ...
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1 vote
1 answer
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Reference request for Fenchel-Rockafellar duality for dual system

In the Fenchel-Rockafellar duality theory, the usual setup contains: $X$-Banach space, $X^*$-continuous dual space of $X$, $\langle x^*, x\rangle := x^*(x)$ where $x^*\in X^*$ and $x\in X$. On the ...
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1 vote
0 answers
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All open convex sets are homeomorphic [duplicate]

Prove that all the open convex (non-empty) subset of $\mathbb{R}^n$ are homeomorphic to each other. First of all I'm not sure if the statements is correct in the first place or not. I sketched a proof,...
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3 votes
0 answers
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Exercise 7, Section 2.2 of Hoffman’s Linear Algebra

Let $W_1$ and $W_2$ be subspaces of vector space $V$ such that the set theoretic union of $W_1$ and $W_2$ is also a subspace. Prove that one of the space $W_i$ is contained in the other. My attempt: ...
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Corollary of Theorem 3, Section 2.2 of Hoffman’s Linear Algebra

If $W_1,…,W_k$ are subspaces of $V$, then the sum $W=W_1 +…+W_k$ is easily seen to be a subspace of $V$ which contains each of the subspace $W_i$, From this it follows, as in the proof of theorem 3, ...
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  • 1,556
1 vote
1 answer
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Is Gram-Schmidt process redudant about square full-rank matrices?

I am trying to grasp the concept of Gram-Schmidt process and I have encountered the following logical difficulty: Given a set of $n$ independent vectors, applying GS algorithm upon this set would ...
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1 answer
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Given a real vector space $V$, each element of the complexification has the form $v+iw$?

Notation: Let $W$ be some complex vector space, then $\widetilde W$ is the associated real vector space. Question: Let $V$ be a real vector space and $(V_\mathrm c,I)$ a complexification of $V$, i.e. $...
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-1 votes
0 answers
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Planes in the space [closed]

How to convince yourself that two planes with proportional coefficients represent the same plane? ax+by+cz+d=0 (ap)x+(bp)y+(cp)z+dp=0--> p(ax+by+cz+d)(1/p)=0(1/p)-->ax+by+cz+d=0 this one is too ...
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0 answers
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Theorem 3, Section 2.2 of Hoffman’s Linear Algebra

In this lecture, professor Artem Chernikov, define linear combination as, $v\in V$ is linear combination of $S$ if $v= a_1\cdot u_1+…+a_n \cdot u_n$ ; $a_i\in F$, $\forall i\in J_n$ and $u_i\in S$, $\...
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0 votes
1 answer
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basis extension intuition

If we have vectors $v_1, v_2, v_3\in \mathbb R^4$ and we want to extend them to a basis of $\mathbb R^4$, why can we take the vectors as rows and then doing rows reductions with Gauß algorithm and ...
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Definition of Subspace of a Vector space

Following is the definition of subspace of a vector space in Hoffman linear algebra book: Let $V$ be a vector space over the field $F$. A subspace of $V$ is a subset $W$ of $V$ which is itself a ...
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