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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55 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} ...
2
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0answers
34 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 ...
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1answer
30 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 \\ ...
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1answer
31 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. ...
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2answers
91 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) $\lambda_1 = 5.9$ $\lambda_2 = 3.5$ ...
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1answer
49 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
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1answer
48 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 ...
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1answer
55 views

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

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 ...
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2answers
44 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
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1answer
21 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
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1answer
52 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
69 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
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1answer
42 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 $K$?
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1answer
122 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
127 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
51 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 ...
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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 ...
1
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1answer
29 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
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1answer
38 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 ...
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1answer
51 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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1answer
30 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 ...
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0answers
36 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 ...
6
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5answers
688 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 ...
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0answers
26 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 ...
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1answer
41 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 = ...
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1answer
37 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
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0answers
40 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
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1answer
56 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
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1answer
33 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 ...
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3answers
59 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
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0answers
60 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 ...
12
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2answers
274 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
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1answer
31 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
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1answer
58 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 ...
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1answer
41 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
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1answer
131 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
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1answer
64 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 ...
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0answers
35 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 ...
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2answers
48 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 ...
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1answer
28 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 = ...
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0answers
30 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
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2answers
333 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 ...
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2answers
76 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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1answer
81 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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2answers
56 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
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3answers
52 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
84 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
0
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1answer
40 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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2answers
26 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
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1answer
15 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 ...