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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70 views

Must a basis for an $n$-dimensional vector space have $n$ vectors?

Does a basis for an $n$-dimensional vector space have to have $n$ vectors? For example, if I form a basis for $\mathbb{R}^n$, do I need at least $n$ vectors in my basis set? In other words, can I ...
0
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1answer
20 views

Finding change of basis matrix when given two bases as a set of matrices

Find the change of basis matrix between the following bases: $\alpha = \left\{ \begin{pmatrix} 1 & 1 \\ -1 & 2 \end{pmatrix}, \begin{pmatrix} 3 & 1 \\ -1 & 2 \end{pmatrix}, ...
1
vote
1answer
44 views

Generating vectors in a non-orthogonal 3D lattice with increasing magnitude

I am trying to build an algorithm to generate a sequence of lattice vectors $\mathbf{v}_n$ in 3D such that: (a) the first vector $|\mathbf{v}_1|$ is the shortest vector of the lattice (b) for all $i ...
2
votes
3answers
21 views

Determining the formula for a linear map

Determine the formula for the following linear map: $L : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ with $L(1,2) = (0,-1)$ and $L(-1,-1) = (2,1)$. Attempt at solution: On the basis of these examples I ...
0
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1answer
20 views

Sesquilinear Forms: Polarization

This thread is only Q&A.* Given a Hilbert space $\mathcal{H}$. Consider the transforms: $$q[\varphi]:=s(\varphi,\varphi)\quad s(\varphi,\psi):=\frac{1}{4}\sum_{\alpha=0\ldots3}i^\alpha ...
0
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0answers
15 views

Linear (In)dependence and other relations

(i) "nontrivial solution" same as "linear dependence" same as "determinant zero" same as "the vectors lie in the same plane". (ii) "trivial solution" same as "linear independence" same as ...
2
votes
1answer
65 views

Sesquilinear Forms: Cauchy-Schwarz

This thread is related: Parallelogram Given a Hilbert space $\mathcal{H}$. Consider a quadratic form: $$q:\mathcal{H}\to\mathbb{C}:\quad q[\lambda\varphi]=|\lambda|^2q[\varphi]$$ Suppose it ...
3
votes
2answers
40 views

For which infinite dimensional real normed linear spaces $X$ , can we say that every infinite dimensional subspace of it is closed in $X$ ?

For which infinite dimensional real normed linear spaces $X$ , can we say that every infinite dimensional subspace of it is closed in $X$ ? Or , does every infinite dimensional normed linear space has ...
0
votes
2answers
41 views

Let V be a vector space of dimension n. Prove that no set of n - 1 vectors can span V.

I'm not sure I understand the question. As far as I understand it when it says vector space of dimension n, it signifies that there will be n amount of vectors; right? So basically it wants you to ...
0
votes
4answers
34 views

Prove or disprove that the set of polynomials of degree greater than or equal to two, along with the zero polynomial is a vector space

This was disproved by giving the example: $$(x^2)+(1+x-x^2)$$ The result is NOT in the set so it's NOT closed under addiction, so NOT a vector space. But I was looking for some prove that doesn't ...
2
votes
1answer
32 views

Suppose $U=Span\{u_{1}, u_{2} \}$ for $u_{1}, u_{2} \in U$ and $V=Span\{ v1, v2\}$ for $v_{1},v_{2} \in V$. Prove that $U+V=Span\{u1,u2,v1,v2\}$.

This is what I have so far, I don't know if this is where I stop or if there is more to prove? $$U+V = (c_{1}u_{1} + c_{2}u_{2}) + (c_{1}v_{1} + c_{2}v_{2}) = c_{1} (u_{1}+v_{1}) + ...
1
vote
3answers
36 views

Proving $\phi: V \rightarrow \mathbb{R}^n$ is linear and finding matrix representation of it

Problem: Let $V$ be a $n$-dimensional vectorspace and let $\beta = \left\{v_1, v_2, \ldots, v_n\right\}$ be a basis for $V$. Prove that the coordinate map $\phi_{\beta} : V \rightarrow \mathbb{R}^n$ ...
0
votes
1answer
24 views

Proof to show that sums of vectors spanning a vector space also span a vector space

Let vectors $v_1, v_2, and v_3$ span a vector space $V$. Show that the vectors $v_1, v_1 + v_2$ and $v_1+ v_2 + v_3$ also span $V$. How would I go about proving this? I understand that I have to show ...
0
votes
1answer
34 views

Definition of the vector cross product

As far as I understand the cross product between two vectors $\mathbf{a},\mathbf{b}\in\mathbb{R}^{3}$ is defined as a vector $\mathbf{c}=\mathbf{a}\times\mathbf{b}$ that is orthogonal to the plane ...
3
votes
3answers
83 views

How does parametrization of the intersection of two surfaces induce a space curve?

Given a two surfaces say: $z=1-y$ and $ x^2+y^2+z^2=1$, we find that they intersect at: $$x^2-2yz=0$$ How is the above a space curve? Is it not just another surface? And why do we need to introduce ...
2
votes
0answers
50 views

Counting chain maps

Let $\mathbb{K}$ be a field and let $C_{\cdot}$ and $K_{\cdot}$ be bounded chain complexes with coefficients in $\mathbb{K}$. Then the set of chain maps $f_{\cdot}:C_{\cdot}\to K_{\cdot}$ is a ...
1
vote
1answer
33 views

Determining kernel and image of linear map

Problem: Which of the following maps are linear? Determine the kernel and the image of the linear maps and check the dimension theorem. Which maps are isomorphisms? 1) $L_1: \mathbb{R} \rightarrow ...
3
votes
3answers
35 views

Vector Valued Functions, Find some value at point

Suppose that $r$ is a vector valued function of $t$. Now, $r_0=\langle 2,2,2\rangle$ and $r_1$ is in the $y,z$ plane. If $r' \times \langle 2,3,4\rangle=0 \forall t$, how can I find what $r_1$ is? I ...
1
vote
1answer
30 views

Proving that $V = U_1 \oplus U_2 \oplus \ldots \oplus U_k$.

Problem: Let $V$ be a vectorspace and $\beta$ a basis for $V$. Now make a partition of $\beta$ in a disjoint union of subsets $\beta_1, \ldots, \beta_k$ and let $U_i = \text{span}(\beta_i)$ for every ...
0
votes
2answers
72 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
24 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
24 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
51 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
42 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
vote
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
39 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 ...
-1
votes
1answer
27 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
191 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
15 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
vote
3answers
35 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
40 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
17 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
20 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
22 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
357 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
15 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
67 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
27 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
30 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
44 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
41 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
17 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
34 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 ...
-2
votes
1answer
42 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
17 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
49 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
22 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 ...