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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1answer
37 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?
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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
44 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 ...
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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?
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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 ...
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 ...
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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 ...
0
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0answers
18 views

What is the dimension of the space $W_2$ if $\text{dim}(W_1+W_2) = 2$, $\text{dim}(W_1 \cap W_2) = 1$ and $W_1 = \langle a_1, a_2 \rangle$?

What is the dimension of the space $W_2$ if $\text{dim}(W_1+W_2) = 2$, $\text{dim}(W_1 \cap W_2) = 1$ and $W_1 = \langle a_1, a_2 \rangle$, where $a_1 = (1,0,0)$, $a_2=(2,0,0)$? Okay so I know and ...
5
votes
5answers
217 views

The definition of span

In Linear Algebra by Friedberg, Insel and Spence, the definition of span (pg-$30$) is given as: Let $S$ be a nonempty subset of a vector space $V$. The span of $S$, denoted by span$(S)$, is the ...
0
votes
0answers
22 views

Dimension of subspace of $(\mathbb{C}^2)^{\otimes n}$

Consider the space $V = (\mathbb{C}^2)^{\otimes n}$ with $n$ even. Let $(v_+, v_-) = ((1,0), (0,1))$ be a basis of $\mathbb{C}^2$. Then the pure tensors $v_{\pm} \otimes \cdots \otimes v_{\pm}$ form a ...
0
votes
1answer
36 views

Find the subspace formed by intersection of given subspaces of $\mathbb{R}^3$

We are given three subspaces of $\mathbb{R}^3$: $W_1 = \{(a_1,a_2,a_3)\in\mathbb{R}^3\mid a_1=3a_2\space$and$\space a_3=-a_2\}$ $W_2 = \{(a_1,a_2,a_3)\in\mathbb{R}^3\mid 2a_1-7a_2+a_3=0\}$ $W_3 = ...
0
votes
1answer
28 views

Finding bases for subspaces of $\mathbb R^3$ and extending them to bases of $\mathbb R^3$

I am given the following question: For each of the sets in Problem 1 which is a subspace of $\mathbb R^3$, find a basis for the subspace, and then extend it to a basis for $\mathbb R^3$. We ...
2
votes
0answers
33 views

Derivative of inv: subset of linear automorphisms

I have no clue how to approach this problem, I've asked for some help from different people, but I have yet to comprehend it. The question is the following, Let $\mathcal L$($\mathbb C$$^n$) denote ...
1
vote
1answer
44 views

What all possible matrix reprsentations a linear operator can have?

$L$ is a linear operator such that $L:V \to V$ where $V$ is a $n$ dimensional hilbert space. If $[L]_{ij}$ is the matrix representation for $L$ in the input and output basis $\{i\}$ and $\{j\}$, then ...
0
votes
1answer
27 views

Proving that vector space is a linearly independent subset of bigger vector space

So I have this problem: Problem $\left\{x,y,z\right\}$ is a linearly independent subset of another vector V, find the constants a and b such that $\left\{x-ay,ay-z,z-by\right\}$ is also a linearly ...
1
vote
3answers
43 views

What is a basis for the vector space $ \Bbb{C}^{n} $ (a complex vector space)?

I know that a basis for $ \Bbb{C} $ is $ \{ 1,i \} $. This set is linearly independent in $ \Bbb{C} $ and spans $ \Bbb{C} $. I think that the dimension of $ \Bbb{C}^{n} $ may be $ 2 n $, but I’m just ...
3
votes
0answers
50 views

Basis of $\mathbb{F}[[x]]$ over $\mathbb{F}$ without AC

Does the ring of formal power series $\mathbb{F}[[x]]$ as a vector space over $\mathbb{F}$ admit a basis without assuming the Axiom of choice, at least in some special cases of $\mathbb{F}$? I'm ...
0
votes
1answer
17 views

An equality in a vector space

Let $V$ be a vector space and $E:V→V$ be a linear transformation. Under what conditions we may have the following: $(E(v)-v)(E(v)-v)^t=(v-E(v))(v-E(v))^t$ (v∈V), where $t$ stands for transpose? ...
11
votes
1answer
214 views

$5$ dimensional space over $\mathbb{R}$

When coming up with a double cover of $SO(5)$, I used conjugation by matrices of the form $$\begin{pmatrix} r & q\\ \overline{q} & r \end{pmatrix}$$ where $r\in\mathbb{R}$ and $q$ is a ...
1
vote
1answer
26 views

Angles between points in $3$D space where the Origin is not the vertex.

Given two points $P_1,P_2$ in $3$D space that are positioned around a third point $M$, how do you calculate the angle between $P_1,M,P_2$. I know there are a few questions on here discussing how ...
1
vote
1answer
41 views

Why should there be a 7-dimensional cross product in the context of exterior algebra?

The three-dimensional cross product can be viewed as the wedge product corresponding to the exterior power $\Lambda^2(\mathbb R^3)$. An explanation that I have come up with for the scarcity of cross ...
0
votes
1answer
26 views

identity operator, direct sums, and projections

Let W be finite-dimensional vector space. Let $P: W\to W$ be a projection. Let U = Range(P) and V=Ker(P) (a) show that P is the identity operator on U. I dont understand the problem ...
0
votes
1answer
51 views

Finding a pair of Orthogonal Vectors

Want: Pair of orthogonal vectors in $R^4$ that are also orthogonal to the vector (1,1,-2,3) My attempt at a solution: I got stuck...
0
votes
1answer
26 views

Let W be the collection of all 2 by 2 symmetric matrices. Describe the orthogonal complement of W. (please)

A matrix is symmetric if $A^T$=A And the standard basis for symmetric matrices is [a,b], [b c] written as rows of a 2x2 matrix (sorry don't know how to make a matrix on this site). My question: How ...
0
votes
0answers
31 views

Vector Space Basis' Proof

Show that : if $B=\{X_1, X_2, \ldots, X_n\}$ and $A= \{ Y_1, Y_2, \ldots, Y_p\}$ are basis' of a vector space $(E, +, \cdot)$ that means $n=p$. I have no idea on how to start this proof, if I can get ...
0
votes
2answers
26 views

How is a Euclidean space a function space?

To be more precise, in what sense is $\mathbb R^N$ a function space? I quote from page number 3, in the first chapter of "Introduction to Hilbert Spaces with Applications" by Debnath and Mikusinski ...
0
votes
1answer
19 views

Let $M =\{ f(x)\in P_3 | \int_0^1f(x)dx = 0\}$ Find basis for M.

Let $M =\{ f(x)\in P_3 | \int_0^1f(x)dx = 0\}$ Find basis for M. solution: $P_3$ is the set of all polynomials of degree strictly less than 3, ($f(x) = a_2x^2+a_1x+a_0$). hence, $\int_0^1f(x)dx = ...
0
votes
0answers
25 views

Prove $K\cap L$ is a subspace of V, but $K\cup L$ is never a subspace.

assume K, L are proper subspaces. Prove $K\cap L$ is a subspace of V, but $K\cup L$ is never a subspace. Solution: if $v_1,v_2\in K$, then $c_1v_1+c_2v_2 \ in K$ [because K is a subspace] if ...
0
votes
2answers
35 views

Prove that if v is orthogonal to u, then it is orthogonal to any scalar multiple of u.

I never understand where to start with proofs, but whenever I see them done I understand them. My attempt: For this one could I just use the property of inner products to prove this? That being ...
0
votes
2answers
33 views

Prove that if u and v are vectors in $\mathbb{R}^n$, then $\langle u,v\rangle =1/4\|u+v\|^2-(1/4\|u-v\|^2)$

I seem to always have troubles when starting proofs. My professor said that the proofs he gave us today are mostly one line proofs, but I just don't know where to start with this one. What I've ...
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votes
1answer
59 views

Finding a pair of orthogonal vectors in $R^4$

Find a pair of orthogonal vectors in $R^4$ that are also orthogonal to the vector (1,1,-2,3). What i have tried so far:
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votes
2answers
32 views

Vector Spaces and Linear Transformations ($T^2 = 0 \iff R(T) \subseteq N(T)$). [duplicate]

Let $V$ be a vector space over a field $F$. Let $T: V\to W$ be a linear transformation. a. Prove that $T^2=0$ if and only if $R(T)$ is contained in $N(T)$. (Here we denote $T^2$ as the linear ...
0
votes
3answers
38 views

find dimension of a vector space

Let A $\begin{pmatrix} 1 & 2 & -1\\ -2 & -4 & 2\\ 0 & 1 & 2\\ \end{pmatrix} $ . Let D = $\{B\in\mathbb{R}^{3x3}| BA = \begin{pmatrix}0 &0&0\\0 &0&0\\0 ...
0
votes
2answers
19 views

how to show $F=\{(a+2b+3c,a-b,-3a+b-2c,2b+2c),\}$ is a subspace?

how to show $F=\{(a+2b+3c,a-b,-3a+b-2c,2b+2c),\}$ is a subspace? i understand closed subspace should be closed under addition and scalar multiplication
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votes
1answer
35 views

How do you find a basis for $\mathbb R^4$ such that it contains specific elements

How do you find a basis for $\mathbb R^4$ such that it contains specific elements: $(2,4,-1,0), (-4,-8,2,1)$
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votes
2answers
15 views

Dimension of an $F$-vector space, compared to dimension over $E:F$

Suppose $E:F$ is a finite field extension of $F$. If $V$ is both an $F$-vector space and an $E$-vector space, then is there any relation between the dimension of $V$ over $F$ and the dimension of $V$ ...
1
vote
1answer
12 views

Invariant subspace (Proof)

How do I prove, that the eigenspaces of $T^n$ are invariant in regard to $T$, assuming T is an endomorphism in a real vector space V $(T: V\rightarrow V)$? That's how I started: Let $E_\lambda$ be ...
0
votes
2answers
35 views

Determining $\dim V$

This is probably a simple question but I'm not sure about the answer. According to Lounesto in Clifford Algebras and Spinors, a vector space of $\dim V=2$ has basis such as \begin{align} ...
0
votes
2answers
42 views

How do I find a relation for these polynomials from a matrix?

I have the following three polynomials: $1 + 2t^2, 4 + t + 5t^2, 3 + 2t$. I need to show that they are linearly dependent in $\mathbb P_2$ (polynomials of degree at most $2$). I put them in a $3x3$ ...
0
votes
1answer
30 views

Suppose $S_1 =\{ u_1 , u_2 \}$ and $S_2 = \{ v_1 , v_2 \}$ are each independent sets of vectors in an n-dimensional vector space V.. [duplicate]

Let us assume that every vector in S_2 is a linear combination of vectors in S_1. Question: Does that mean that S_1 and S_2 are bases for the same subspace of V? I know that the answer to this ...
0
votes
0answers
35 views

show linear transformation bijective

Can you please help me prove this? Let $T:\mathbb{R}^7\to\mathbb{R}^7$ be a linear transformation such that 9 is an eigenvalue of $T$ and $dim(E_9)=6$ Prove that either T-4I or T-5I is a bijection ...
0
votes
2answers
46 views

Suppose $S_1 =\{ u_1 , u_2 \}$ and $S_2 = \{ v_1 , v_2 \}$ are each independent sets of vectors in an n-dimensional vector space V.

Let us assume that every vector in $S_2$ is a linear combination of vectors in $S_1$. Question: Does that mean that $S_1$ and $S_2$ are bases for the same subspace of $V$? I know that the answer to ...
1
vote
3answers
61 views

Determine whether the set $\{v_1 + v_2 - v_3, 2v_1 + 2v_3, -v_1 + v_2 - 3v_3\}$ is linearly dependent or independent.

We had a question on our last test that was very similar to this and I only got $2$ points of $6$ and I want to make sure I do it right this time. Here's my solution to that one: Let $v_1, v_2,$ and ...
0
votes
4answers
39 views

Let V be a vector space and W a subset of V. Suppose zero is in W and W is closed under addition. Is W a subspace of V?

I know that the answer to this question is No. My question is why is the answer no? What's missing? if possible give a specific example of both V and W such that W satisfies above conditoins but it ...