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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2answers
24 views

Prove $\exists$ $v \in V$ so that $(v , f(v))$ is a basis of $V$

maybe you guys can help me with this one. Let's say we have a vector space $V$ with $dim(V) = 2$ and we have a linear map $f : V \rightarrow V$ with $f^2 := f \circ f = 0$ ...
0
votes
2answers
40 views

Consider the vector space V = {(a, 1 + a) | a ∈ R} with irregular definitions of addition and multiplication

with addition and scalar multiplication defined by (a, 1 + a) ⊕ (b, 1 + b) = (a + b, 1 + a + b) k '*' (a, 1 + a) = (ka, 1 + ka), k ∈ R find a basis for V. I started off with taking the general ...
0
votes
1answer
23 views

Determine the dimension of $U+W$ and of $U \cap W$. Which sums are direct sums?

Problem: Determine the dimension of the sum $U + W$ and of the intersection $U \cap W$ of the following subspaces $U$ and $W$. Which sums are direct sums? 1) $U = \text{span}\left\{(1,1,1)\right\}$ ...
1
vote
2answers
20 views

Are the difference of two vectors orthogonal if the angle between the two vectors approaches 0? (Attempted proof)

Suppose that $\vec{a}=(x,y), \vec{a`}=(x', y'), \Delta \vec{a} = (x'-x, y'-y), \theta \rightarrow 0$ where $\theta$ is the angle between $\vec{a}$ and $\vec{a'},$ and the magnitudes are equal, $a=a'$ ...
0
votes
3answers
30 views

Show that $\ker \hat{T} = \text{ann}(\text{range } T)$

This is an old exam problem: Let $V$ and $W$ be finite dimensional vector spaces over a field $F$ and let $T: V \to W$ be a linear transformation. Define $\hat{T}: W^* \to V^*$ by ...
2
votes
1answer
43 views

Countable vector space of continuous functions over a compact metric space

In a proof of a specific theorem, the following is stated: ($\Omega$ is assumed to be a compact metric space) "Let $H \subset C(\Omega)$ be a countable vector space over $\mathbb{Q}$ which is closed ...
2
votes
1answer
39 views

Finding a basis for $V, W, V+W$ and $V \cap W$

Problem: Let \begin{align*} V = \left\{(x,y,z,u) \in \mathbb{R}^4 \mid y+z+u = 0 \right\} \end{align*} and \begin{align*} W = \left\{(x,y,z,u) \in \mathbb{R}^4 \mid x+y = 0, z = 2u \right\} ...
1
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1answer
11 views

Use of GS before projecting a vector onto a plane

I need help with the following exercise: Given the vectors $u_1 = (2,-1,2), u_2 = (1,2,1), u_3 = (-2,3,3)$, what is the projection of $u_3$ onto the plane spanned by $u_1$ and $u_2$. I'm not sure if ...
1
vote
2answers
37 views

Number of vectors over a finite field that are linearily independent to a subspace

let $S$ be a vector space over a finite field of size $q$ and let $T$ be a subspace of $S$. I am looking for a formula or an algorithm to compute the number of vectors from $S$ that are independent ...
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1answer
25 views

Prob. 3, Sec. 4.2 in Erwin Kreyszig's Functional Analysis: How to show that $\lim\sup$ is sublinear?

Let's consider the real space $\ell^\infty$ of all bounded sequences of real numbers. Let $p \colon \ell^\infty \to \mathbb{R}$ be defined by $$p(x) \colon= \lim\sup_{n \to \infty} \xi_n \ \mbox{ for ...
6
votes
1answer
184 views

definition of ordered vector space

An ordered vector space is the pair $(V , \leq)$ where it satisfies the following: For all $x,y,z \in V, \lambda \geq 0$, i) $x \leq y \Rightarrow x+z \leq y+z$ ii) $x \leq y \Rightarrow \lambda ...
0
votes
0answers
14 views

Why do we need to worry about convergence in $\mathbb{R^Z}$ if each $\mathbf{e}_i$ are already pairwise linearly independent?

(Note: as pointed out by some users in related questions, the $\mathbb{R^\infty}$ in the link turns out to be $\mathbb{R^Z}$) Once again, a question inspired from reading this An excerpt One ...
1
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3answers
34 views

How to determine a basis and the dimension for this vectorspace?

Determine a basis and the dimension for the following vectorspace: \begin{align*} W = \left\{A \in \mathbb{R}^{3 \times 3} \mid A \ \text{is a diagonal matrix and} \ \sum_{i=1}^3 A_{ii} = 0\right\} ...
1
vote
1answer
38 views

Checking if a matrix is in the span of other matrices

Problem: Expand the following set matrices \begin{align*} \left\{ \begin{pmatrix} 1 & -1 \\ 2 & 3 \end{pmatrix}, \begin{pmatrix} 2 & 1 \\ -1 & 4 \end{pmatrix}, \begin{pmatrix} 0 & ...
0
votes
0answers
14 views

Continuous action on tensor product

Let $G$ be a profinite group and $V,W$ be $k$-vector spaces with discrete topology. Suppose $G$ acts continuously on $V$ and $W$, we extend the action of $G$ to $V \otimes_k W$ by defining on simple ...
2
votes
1answer
14 views

How to see transformations on polytopes?

I have a polytope in six dimension with extreme points $(1,0,0,0,0,0)$ $(0,1,0,0,0,0)$ $(0,0,1,0,0,0)$ $(1,1,0,1,0,0)$ $(1,0,1,0,1,0)$ $(0,1,1,0,0,1)$ $(1,1,1,1,1,1)$ $(0,0,0,0,0,0)$ Each of the ...
1
vote
1answer
21 views

Equivalence for direct sum of vector subspaces

I have a lot of problems proving the following statement. Let $V$ be a vector space. Let $W,K\leq V,$ where $\leq$ denotes vector subspaces. $W\bigoplus K=V$ $\iff$ $K\leq V$ and is least ...
1
vote
1answer
14 views

How to find the facet inequalities for Bell-Wigner polytope?

The Bell-Wigner polytope has the following extreme points $(1,0,0,0,0,0)$ $(0,1,0,0,0,0)$ $(0,0,1,0,0,0)$ $(1,1,0,1,0,0)$ $(1,0,1,0,1,0)$ $(0,1,1,0,0,1)$ $(1,1,1,1,1,1)$ $(0,0,0,0,0,0)$ I checked ...
5
votes
2answers
351 views

An example for infinite dimensional vector space with Hamel dimension smaller than $\operatorname{card} F$

What will be the example for a vector space(infinite dimensional) over a field where Hamel basis has strictly smaller cardinality than that of field? It is not possible in a Hilbert Space (over R or ...
0
votes
0answers
14 views

If $S$ and $T$ are closed vector subspaces then $S+T$ is closed [duplicate]

Let $V$ be a Banach normed space, $S,T \subset V$ be closed vector subspaces. Assume $\operatorname{dim}(T)<\infty$. Show that $S+T$ is closed. So I encountered this problem trying to use ...
1
vote
2answers
62 views

Is the Cauchy-Schwarz inequality ever used in Physics?

Given that Physics uses vectors extensively, and that the most natural setting for the Cauchy-Schwarz inequality is a vector space, the question naturally arises: Is the Cauchy-Schwarz inequality ever ...
0
votes
1answer
26 views

Method for finding intersection between two basis

What is the general way to find a basis for the intersection of two sub spaces? There's the method that use the fact that if we take some vector $v\in V$ and $v\in U$ then every linear combination ...
0
votes
0answers
29 views

An exercise question on Hoffman's Linear Algebra

Is the vector (3,-1,0,-1) in the subspace of $R^5$ spanned by the vectors (2,-1,3,2), (-1,1,1,-3), (1,1,9,-5)? I think these vectors all live in $R^4$ instead of $R^5$ so they the answer is no, but ...
3
votes
1answer
38 views

Why does a differential form represent a vector field?

I'm trying to learn the Divergence/Stoke's theorem and I can't wrap my head around the meaning of a differential form in this context. What does it mean that a differential form represents a vector ...
0
votes
1answer
40 views

Losing a dimension when finding intersection between subspaces

Let $F=\mathbb Z_3, V=F^4$. Let $U=sp\{(1,0,0,0),(1,0,1,0),(0,1,1,1) \} \\W=sp\{(0,0,1,0),(-1,1,0,1),(1,1,1,1) \}$ Find $dim (U\cap W)$ we have $v\in U \text{ and } v\in W$ so $v=v$ ...
0
votes
1answer
16 views

Short question about modulo space $\mathbb Z^n_p$ and the zero vector

Say we have a vector in $\mathbb Z^3_5$: $v= (1,2,0)$ it looks like it isn't the zero vector but if we multiply it by a scalar: $5v=(5,10,0)\overset{mod5}=(0,0,0)$ so now it is the zero vector and we ...
2
votes
1answer
45 views

A linear algebra question

$V$ is a vector space with finite dimension. Let $f_1, \ldots,f_m\in\operatorname{End}(V)$ be linear maps of $V$ to itself. Suppose that $V=\ker(f_1)+\ldots+\ker(f_m)$. Show that there are $g_1, ...
0
votes
1answer
33 views

A question on vector subspace [duplicate]

Let $V$ be the vector space of all functions $f \colon \mathbb{R} \to \mathbb{R}$ over $\mathbb{R}$, is the set of functions which are continuous a subspace? I think if you add functions which are ...
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votes
1answer
36 views

A excercise problem on Hoffman Linear Algebra

Let $V$ be the vector space over $\mathbb R$ of all functions $f :\mathbb R \to\mathbb R$, then identify if the following is a subspace of $V$: All $f \in V$ such that $f(x^2)=f(x)^2$ While I ...
1
vote
2answers
16 views

rotating linear dependent vectors in space

I'm not quite sure how to write this succinctly with mathematical symbols, so I just had to write it out in english. Any edit to suggest how to write it in mathematical form would be appreciated even ...
2
votes
1answer
45 views

Prove that the product of 2 vectors Normally distributed converges for large dimensions to the full zero matrix

Let $\mathbf{x}, \mathbf{y}$ $\in C^{M \times 1}$ are two i.i.d. vectors with distribution $\mathcal{CN(0,1)}$. How we can prove by the strong law of large numbers that: $\lim_{M\rightarrow \infty} ...
0
votes
1answer
18 views

About finding intersection between two vector spaces

Let $W=sp \{e_1,e_2,e_3,e_4\}, U= sp\{(1,-2,1,0),(0,3,-1,1)\}$ be vector spaces both are linearly independent. Show that $U\cap W = sp\{(3,0,1,2)\}$. I know that $\dim U\cap W =1$. Now ...
0
votes
1answer
15 views

Subvector and related subspace

This might be easier than I think, but I got stuck. Assume a vector $y=[y_1,\ldots,y_n]\in Y$, where $Y$ is a convex polyhedron. Assume a $k$-dimensional subvector of $y$, namely ...
2
votes
1answer
52 views

Vector Calculus Operator $\vec{u} \cdot \nabla$

I just want to double check on this operator and it's properties. It pops up in fluid mechanics often and I just want to be sure about my understanding: 1) $$(\vec u \cdot \nabla)\vec u$$ Is this ...
1
vote
2answers
44 views

How to check is two subspaces are the same.

Suppose I have some $N$ dimensional real vector space and two $M<N$ dimensional subspaces of that, and say I know one set a basis vectors for each: ${v_i}$ where $i=1,2,...,M$ and ${w_i}$ where ...
2
votes
1answer
59 views

$\operatorname{span}(x^0, x^1, x^2,\cdots)$ and the vector space of all real valued continuous functions on $\Bbb R$

Let $p_n(x)=x^n$ for $x\in\Bbb R$ and let $\mathcal P=span\{p_0,p_1,p_2,\cdots\}$ . Then $\mathcal P$ is the vector space of all real valued continuous functions on $\Bbb R$. $\mathcal P$ ...
0
votes
1answer
33 views

Hilbert space isometric to a subspace of its dual

Let $\cal H$ be a Hilbert space, and let $\cal H^\ast$ be its dual (of the continuous functionals). If $\cal H$ is a real vector space, I can define: $$\begin{align}\Phi\colon\, &{\cal H} \to ...
6
votes
1answer
126 views

Prove $AB=I$ implies $BA=I$ using Fitting's Lemma

I know this question has already been asked, but I need a proof that for $A,B \in M_n(K)$, $$AB=I_n \Rightarrow BA=I_n$$ using Fitting's lemma. I thought of using the fact that $K^n$ is a ...
1
vote
2answers
42 views

Is 'basis times square matrix' a new basis?

Suppose we have a vector space $V = (K, +, \cdot)$. Let $B$ be a basis of V. Now we take an arbitrary square matrix $S \neq 0$. $BS$ is just a linear combination of B. Thus $BS$ should be a new ...
1
vote
1answer
19 views

Vector space generated by set intersection

Again, I've come across a simple task, but being new to linear algebra, I wish not yet to question my textbook's author credibility. In vector space $R^4$ , two subspaces $W_1$ and $W_2$ are generated ...
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votes
0answers
38 views

Finding a basis for $R^2$ with some constraints

Find a Basis $B$ of $R^2$ s.t. (1) $\left(\begin{array}{c}1 \\2\end{array}\right)_B = \left(\begin{array}{c}3 \\5\end{array}\right)$, and (2) $\left(\begin{array}{c}3 \\4\end{array}\right)_B = ...
1
vote
1answer
15 views

Linear mapping from square matrix vector space to polynomial vector space

Let $M_2(\mathbb{R})$ be a vector space of all 2 dimensional square matrices and $P$ be the space of all $2^{nd}$ degree polynomials. Suppose we have a linear mapping defined as ( from here on, $0$ ...
1
vote
4answers
54 views

Prove that $\{x^2+x , x^2-1, x+1\}$ generates vector space of $2^{nd}$ degree polynomials

I am quite new to linear algebra and am having some trouble with the abstractness of some it's parts. For example, this task seems quite simple and as if there's no need to do any proving, but to ...
4
votes
2answers
73 views

Proof that $\mathbb{R}^+$ is a vector space

I was doing some beginner linear algebra tasks and stumbled upon this one: Proove that $\mathbb{R}^+$ is a vector space over field $\mathbb{R}$ with binary operations defined as $a+b = ab$ (where ...
0
votes
1answer
17 views

Inconsistency of column representation with orthogonality of vectors

Let's say I have two vectors $v_{1}$ and $v_{2}$ which form a basis for $\mathbb{R}^2$. Any vector $v$ in $\mathbb{R}^2$ can be represented as $$v = av_{1} + bv_{2}$$ for some $a,b \in \mathbb{R}^2$. ...
1
vote
1answer
24 views

Proving parallel planes in $\mathbb{R}^4$

Given two planes in $\mathbb{R}^4$ (or perhaps higher dimensions) in parametric form, what ways are there to prove that they are parallel (or not parallel)? A friend suggested equating the spans and ...
0
votes
2answers
17 views

Is equation of a hyperplane fixed?

If I have a $n$ dimensional vector space ( real components ) then a hyperplane will be $n-1$ dimensional. The equation of a hyperplane is defined as $\vec{n}.\vec{x}=\vec{n}.\vec{x_0}$ ( if I am not ...
4
votes
2answers
103 views

Prove that $\dim(V)$ is even

Let $V$ be a finite dimensional vector space. Let $A_1,A_2: V\rightarrow V$ be commuting linear operators such that $A_1+A_2=-I$ where $I$ is the identity operator. Also $A_1,A_2$ have no negative ...
5
votes
1answer
33 views

Subspaces of $\Bbb R^n$ containing vectors whose coordinates satisfy prescribed inequalities

For any integer $n\ge2$, the vector space $\Bbb R^n$ is divided into $n!$ "wedges" by prescribing which coordinate is largest, second-largest, etc. One such wedge is $$\{(x_1,\dots,x_n)\in\Bbb ...
2
votes
2answers
75 views

Regular Quadratic Space - isotrope vector

I am currently trying to solve the following exercise: Show that every regular quadratic space of finite dimension $E$ that contains at least one isotrope vector, has a basis consisting only of ...