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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5answers
592 views

Can the integers be made into a vector space over any Finite Field?

Given a Finite Field $F$, can the the abelian group $\mathbb Z$ be made into a vector space over $F$ without changing the additive structure of $\mathbb Z$? This seems like it shouldn't be ...
2
votes
1answer
20 views

Proving surjectivity and injectivity of two transformations, knowing the rank of their composition.

I have got another question concerning linear algebra. The excercise is: Let ...
0
votes
2answers
38 views

Proving that there is a unique linear map such that $T(u_i)=v_i$.

I have a problem with understanding of a rather simple concept in linear algebra. I have seen in a book, a following question: Suppose $U,V$ are vector spaces over $K$ and $u_1,\dots,u_n$ is a ...
0
votes
0answers
20 views

Necessity of continuity in Topological Vector Space

In the notion of a topological vector space, we define such as a vector space $X$ (over a field $\mathbb{K}$) with topology $\mathscr{T}$ such that $$\iota_+: (X,\mathscr{T}) \times (X, \mathscr{T}) ...
2
votes
2answers
34 views

Does a direct sum decomposition of an infinite-dimensional vector space require Zorn's lemma?

Let $V$ be an infinite-dimensional vector space and $V'\subset V$ a subspace. Does it require Zorn's lemma to write $V=V'\oplus V''$ for some other subspace $V''\subset V$?
0
votes
1answer
33 views

Proving that a set is a subspace of a given space.

I have encountered this question and I am wondering whether my thinking is correct. We have vectors $u_1, u_2, \dots, u_n$ as elements of the space $V$. We have also got a set $U$ is the set of all ...
2
votes
1answer
22 views

Prove relationship regarding the scalar product

For 2 vectors $a,b$ $\in \mathbb{R}^n$ and all entries in the vectors are $\geq 1$ is the following relationship true ? : $\langle a,b \rangle$ $ \leq$ $0.5 \langle a,a \rangle + 0.5 \langle b,b ...
0
votes
2answers
18 views

Determining if vector space holds

Let A be a particular vector in $\Bbb R$2x2. Determine whether the following is a subspace of $\Bbb R$2x2: S = {B ∈ $\Bbb R$2x2 | AB + B = O} I have two ideas on how to approach this for scalar ...
1
vote
1answer
24 views

Factor Modules/Vector Spaces and its basis with canonical mappings

I'm having trouble with factor modules now. Well, specifically the following question from a past paper. Q. T=$R^2$ i.e. the real plane, and define $f:T$->$T$ with respect to the standard basis which ...
2
votes
0answers
31 views

a norm is symmetric if and only if it is unitarily invariant

how can I prove this : A norm on $\mathcal{M}_n(\mathbb{C})$ is symmetric if and only if it is unitarily invariant ? My attempt I know that a symmetric norm is a norm which verifies : $$N(ABC)\leq ...
0
votes
1answer
29 views

Is a linear span of finite set from a finite dimensional space topologically closed?

Let $S=\{x_1,\ldots,x_m\} \subset \mathbb{C}^n $ is it true that: $$ Span (S) = \overline{Span (S)} $$ Must we assume both of the following assumptions? or one of them will be enough? The spanning ...
1
vote
0answers
21 views

Scalar Triple Product

Prove that if $\:$$\vec{a}$, $\vec{b}$, $\vec{c}$ and $\vec{r}$ are any four vectors and if $[\vec{x} \: \vec{y} \: \vec{z}]$ is Scalar Triple product, Then $$[\vec{a} \: \vec{b} \: ...
0
votes
1answer
40 views

What separates a subspace from the entire vector space it is a subspace of?

This is more of a philosophical question about what differs the definitions of a vector space and a subspace. In looking at subspaces of a vector space V, we have one such approach through linear ...
1
vote
2answers
29 views

Invariant Subspace containing linear combination of eigenvectors

Let $$T:V\to V$$ be a linear transformation. Suppose that $v_1, v_2, \cdots, v_k \in V$ are eigenvectors of $T$ that correspond to distinct eigenvalues. Assume that $W$ is a $T$-invariant subspace of ...
2
votes
1answer
36 views

Is a Linear Transformation a Vector Space Homomorphism?

I see the terms linear transformation and (vector space) homomorphism used more or less interchangeably, and the set (space) of linear transformations from V to W referred to as Hom(V, W) or ...
1
vote
0answers
56 views

Conjugation in the complexification of a vector space switches its type

Let $V$ be a real vector space with an almost complex structure $J$ and consider its complexification $V^\mathbb{C}$ where we extend $\mathbb{C}$-linearly the linear maos of $V$, in particular $J$. In ...
3
votes
1answer
22 views

Additive closure in the set of all functions $f \in \mathcal{F}(S,\mathbb{F})$ such that $f(s)=0$ for all but a finite number of elements of $S$

Let $S$ be a nonempty set and $\mathbb{F}$ a field. Let $\mathcal{C}(S,\mathbb{F})$ denote the set of all functions $f \in \mathcal{F}(S,\mathbb{F})$ such that $f(s)=0$ for all but a finite number of ...
2
votes
1answer
27 views

Linearly Independent Linear Transformations

I am currently studying some theories of single linear transformations. I feels like I understant 99% of it, but there is still one thing that I have not been able to resolve. My book explains it by ...
0
votes
1answer
23 views

A transformation problem in linear algebra

any idea how to approach this? given: $$ T:R^3\rightarrow R^3 $$ $$ T(a,b,c)=(b,c,a) $$ $$ B={(0,1,0),(1,0,0),(0,0,1)} $$ find $[3T^{n+2}+3T^{n+1}+3T^n]_B$ for every n
0
votes
1answer
15 views

control system satellite - bdot

I am in charge with developing a control system for a satellite. But there are a few things, about the mathematics, which are not so clear: $\dot{B}_{x}$ rate of change of $B_x$ with respect to time ...
-3
votes
0answers
31 views

Linear transformation matrix.

I can't figure out how to do this? I have no idea how to start part (i).? Any help to get me started?
0
votes
1answer
23 views

Linear Transformation, Nullspace, and Range

Let $V$ be a finite-dimensional vector space over a field $F$. Suppose $P:V\to V$ is a linear transformation such that $P^2=P$. Such a linear transformation is called a projection. Prove that, for ...
0
votes
1answer
32 views

Suppose $V$ is finite-dimensional, Will $L(V,W)$ be infinite-dimensional?

Suppose V is finite-dimensional with $dimV > 0$, and suppose W is infinite-dimensional. Will the linear map from V to W ,denoted $L(V,W)$ ,be infinite-dimensional?
0
votes
1answer
23 views

Linear transformations and matrix basis.

Hello I am currently stuck on this problem: I have no idea how to start part (i). Any help to get me started?
0
votes
0answers
20 views

Finding the basis one forms (covectors) corresponding to a particular formulation of basis vectors

This formulation of the basis may be wrong, or I may be missing something, but I can't see a way to formulate the covectors this particular basis: \begin{align} \vec{e}_0 &= \vec{x} + \vec{y} ...
1
vote
0answers
15 views

The closure of a subspace of a normed vector space is a subspace

This is a self-study problem (Folland Real Analysis exercise 5.5). If $\mathcal{X}$ is a normed vector space, the closure of any subspace of $\mathcal{X}$ is a subspace. My attempt: It is ...
1
vote
1answer
25 views

Finding Eigenvectors when we have lots of zeroes

\begin{array}{cc} 0 & 0 \\ 0 & 8 \\ \end{array} I have $\lambda_1=8$ and $\lambda_2=0$ but cannot find $V_1$ or $V_2$ I try \begin{array}{cc} \lambda & 0 \\ 0 & 8-\lambda \\ ...
1
vote
1answer
24 views

A vector or a set of vectors

Eigenvalue problem: Ax = $\lambda$x Why is x defined as a single vector (eigenvector)? I would think of it rather as a set of three vectors, each in a different dimension. ...
1
vote
2answers
85 views

I have the Eigenvalues, how do I get Eigenvectors?

My matrix is \begin{array}{ccc} 3 & 4 & 5 \\ -2 & 7 & 3 \\ 5 & -8 & -3 \end{array} Through the rule of Sarrus, I know (approximately) $\lambda_1 = 5.9$ $\lambda_2 = 3.5$ ...
0
votes
1answer
18 views

incomplete vector space of continously differentiable functions

Consider the vector space $C^1[a, b] := \{f: [a, b] \to \mathbb{C} \space |\space f$ continuously differentiable$\}$. I now want to show that ($C^1[-1, 1]$, $||.||_\infty)$ is not complete (using ...
2
votes
1answer
29 views

Is there anyway to check I have an an orthogonal and/or orthonormal basis?

I'm reading about Gram-Schmidt procedure in 3 dimensions. From what I understand the idea is to "fix" one of the vectors and alter the other 2 so they are all perpendicular. So say i have three ...
0
votes
0answers
20 views

Prove $U_1⊕…⊕U_m$ is finite-dimensional and $dim U_1⊕…⊕U_m = dimU_1+…+dimU_m$

Suppose $U_1,...,U_m$ are finite-dimensional subspaces of V such that $U_1+...+U_m$ is a direct sum. How to apply $dim(U+V)=dimU+dimV-dim(U∩V)$ to more than 2 subspaces? Please help me with a rigorous ...
-3
votes
1answer
35 views

Sum of two vector subspaces [closed]

V and W are vector subspaces $$ V = \left\{(x, y, z) \in \mathbb{R}^3, x + 2y -z = 0\right\} $$ $$ W = \left\{(x, x, x), x \in \mathbb{R}\right\} $$ Calculate V + W
-1
votes
1answer
38 views

let $α, β, γ, δ$ be endomorphisms such that $α − β$ and $α + β$ are automorphisms. Show that exist $ϕ$, $ψ$ such that $ϕα + ψβ = γ$, $ψα + ϕβ = δ$.

I need some help with this problem: Let $F$ be a field of characteristic other than 2. Let $V$ be a vector space over $F$ and let $α, β, γ, δ$ be endomorphisms of $V$ satisfying the condition that $α ...
0
votes
2answers
40 views

Find the matrix of a linear map $V \to V$, where $V \cong \Bbb R^3$

I am quite new to linear maps, and I have missed a lecture, and for these reasons I am little bit struggling with the exercises I have to do. I have the following problem: Let $V$ be a ...
2
votes
1answer
10 views

Is there a way to recover the sum of a vector coefficients?

Assuming an inner product between two vectors $\mathbb{a}$ and $\mathbb{b}$, $\langle \mathbb{a}\cdot \mathbb{b}\rangle$=v. Is there a way by knowing v and $\sum{\mathbb{b}}_i$ to obtain ...
2
votes
1answer
36 views

Isomorphisms between infinite dimentional spaces

Let $V$ be an infinite dimensional vector space. Can we find an isomorphism between $V$ and $V \oplus V$. If the answer is positive then how this isomorphism can be constructed?
0
votes
1answer
22 views

Transform vector from xy plane to “another vector's plane”

I've not had linear algebra yet, so bear with me if I write something weird. Given vector A and B as shown above, how do I transform vector B so that one of its components is parallel to A, and the ...
1
vote
1answer
40 views

Does $\text{End}_K(K^n) \cong \text{Mat}(n\times n, K)$?

Let $K$ be a field, $K^n$ a vector space over $K$. Is the following true? $\text{End}_K(K^n) \cong \text{Mat}(n\times n, K)$ Does this change if $K$ is a ring, and $K^n$ a module over $K$?
1
vote
1answer
43 views

Zorn's Lemma's chain condition

Zorn's Lemma requires that every chain in a partially ordered set $X$ has an upper bound. In this article Gowers uses Zorn's Lemma to find a maximal linearly independent (over $\mathbb{Q}$) subset of ...
0
votes
0answers
25 views

Absorbing sets on a vector space

The following definition for absorbing set is base in here. With this definition, is it true that finite intersection of absorbing sets is also absorbing?
0
votes
2answers
43 views

Why is the inclusion of the $ 0 $-vector part of the definition of a subspace?

I am not seeing why a subspace must include $ 0 $. From what I am told, this inclusion means that the subspace is not “empty”, but I cannot see how the inclusion of $ 0 $ does this. For instance, can ...
-1
votes
1answer
35 views

linear algebra: deciding if this is a vector space over R? [closed]

Checking if this is a vector space over R The set of continuous real-valued functions on the interval [0,1] s.t. f(0)=0 and f(1)=1?
0
votes
1answer
27 views

$V$ is a vector space not finitely generated, $σ_0\neq α ∈ End(V )$, $A = \{β ∈ End(V ) \mid αβ = σ_1\}$. Show that $A$ is infinite or has one element

I need some help with this problem: Let $V$ be a vector space over a field $F$ which is not finitely generated, and let $σ_0\neq α ∈ End(V )$. Set $A = \{β ∈ End(V ) \mid αβ = σ_1\}$. Show that if ...
1
vote
2answers
31 views

Correct to write $\vec{F}:\mathbb{R}^3\rightarrow\mathbb{R}^3$?

Suppose I have some vector field \begin{align} \vec{F}\left(x\left(t\right),y\left(t\right),z\left(t\right)\right)&=G\textbf{i}+H\textbf{j}+T\textbf{k}.\tag{1} \end{align} Would it be correct for ...
0
votes
1answer
21 views

Let $V$ be as vector space over a field $F$ and let $α, β, γ ∈ \operatorname{End}(V )$ satisfy $αβ = σ_1 = αγ$. Show that $βγ \neq γβ$.

I need some help with this problem please: Let $V$ be as vector space over a field $F$ and let $α, β, γ ∈ \operatorname{End}(V )$ satisfy $αβ = σ_1 = αγ$. Show that $βγ \neq γβ$. $σ_1$ is the ...
0
votes
1answer
34 views

let $α, β ∈\operatorname{ End}(V )$ satisfy $3α^3 + 7α^2 − 2αβ + 4α − σ_1 = σ_0$. Show that $αβ = βα$.

I need some help with this problem please: Let $V$ be a vector space finitely generated over $\mathbb Q$ and let $α, β ∈ \operatorname{ End}(V )$ satisfy $3α^3 + 7α^2 − 2αβ + 4α − σ_1 = σ_0$. Show ...
1
vote
1answer
22 views

Not subspace, but closed under addition and under taking additive inverses?

My linear algebra book (Linear Algebra Done Right by Sheldon Axler) has the following problem as exercise 1.6: Give an example of a nonempty subset $U$ of $\mathbb{R}^2$ such that $U$ is closed ...
0
votes
0answers
6 views

$α_{ij}α_{kh} = α_{ih}$ if $j = k$, $σ_0$ otherwise . There exists a basis of $V$ such that $α_{jk}(v_i) = v_j$ if $i = k$, $0_V$ otherwise

I need some help with this problem please: Let $V$ be a vector space of finite dimension $n$ over a field $F$ and let $\{α_{ij} | 1 ≤ i, j ≤ n\}$ be a collection of endomorphisms of $V$, not all of ...
0
votes
1answer
16 views

Complex Vector spaces inner product superposition axiom

In my studies of Quantum mechanics, the following problem with complex vector spaces has come up, specifically as regards the inner product in such a space. Now in Shankars "Principles of Quantum ...