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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234 views

Orthogonal Complements and Subspaces Proof

I'm having a little difficulty understanding the proof for orthogonal complements. I kind of understand orthogonal complements, but I cannot seem to find a logic to this. I'm trying to follow along ...
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1answer
11 views

Continious subbundle

Let $W$ be a vector bundle with base $\Omega$ and projection $p$. A continious subbundle of $W$ is a subset $W_0$ of $W$ such that $p|W_0$ defines a vector bundle over $\Omega$. Now here ...
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0answers
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What is meant by $\Omega=\text{cls}\left\{a_{\tau}|\tau\in\mathbb{R}\right\}$?

Let $\mathcal{B}=\left\{b\colon\mathbb{R}\to M^n| b \text{ is uniformly bounded and uniformly continious}\right\}$; give $\mathcal{B}$ the compact-open topology. It can be shown that the map ...
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3answers
33 views

A direct proof on if $(X, ||\cdot ||)$ is a normed vector space and $Y\subset X$, with $Y$ having finite dimension, then $Y$ is closed.

I am trying to produce a direct proof on the statement mentioned above. The field I am working in is $\mathbb{R}$. My proof outline goes as following: If $Y$ is finite-dimensional, there exists a ...
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0answers
10 views

dim of $\Bbb R^3 \otimes_\Bbb R \Bbb C$ when considering as a $\Bbb C$-vector space

I'm looking at Sergei Winitzki's Linear Algebra via Exterior Products, and he has a question on tensor products. Firstly we construct the real vector space $\Bbb R^3 \otimes_\Bbb R \Bbb C$ which is ...
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1answer
37 views

the set of points equidistant from $ u $ and $v$ form a line.

Let $u$ and $v$ be two vectors in $ \mathbb{R}^2 $ with the standard norm. Show that the set of points equidistant from $ u $ and $v$ form a line. I show that if $x$ is equidistant from $u$ and $v$, ...
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0answers
12 views

Reducing Subspaces: Nonexample?

Given a Hilbert space $\mathcal{H}$. Consider an operator $T:\mathcal{D}(T)\to\mathcal{H}$. Suppose there exists a closed subspace $Z\leq\mathcal{H}$: $$TZ\subseteq Z,TZ^\perp\subseteq Z^\perp$$ ...
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0answers
18 views

Problem in proving norm

I have a linear space V that includes the continuous functions from [-$\pi, \pi$] to the complex set C, of the form: $$f(x) = a\cos(t)+b\sin(t) $$ where $a$ and $b$ are complex numbers. I want to ...
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1answer
22 views

Vector spaces and direct sums

The map that was constructed in lectures is: $V,W$ subspaces of $U$. $f\colon V \oplus W \to U$ by the formula: $f((v,w))=v+w$ for $v$ in $V$, $w$ in $W$ Is it correct to generalise this to, ...
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2answers
31 views

Is union of two subspace a subspace too? [duplicate]

Assume that $W$ and $V$ are two subspace of $X$. Is their union a subspace of $X$ too? I think it is not true unless under certain conditions but I do not know what conditions...
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1answer
35 views

Proof vector x + ⃗y = ⃗x + ⃗z then ⃗y = ⃗z

Let ⃗x, ⃗y and ⃗z be vectors in a vector space V . Prove that if ⃗x + ⃗y = ⃗x + ⃗z then ⃗y = ⃗z. No idea how to start.
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1answer
16 views

Prove that the linear map of the basis $V$ is a spanning set of the image of $f$

Suppose that $f:V\rightarrow W$ is a linear map of finite-dimensional vector spaces and that $S=\{v_1,v_2,...,v_n\}$ is a basis for $V$. Prove that $\{f(v_1),f(v_2),...,f(v_n)$} is a spanning set ...
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3answers
30 views

Is set of all contiuous functions subspace?

This is one of the problems from the book: Hoffman and Kunze, chapter: Vector Spaces Let V be the (real) vector space of all functions f from R into R. Is the set of all f which are continuous, ...
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0answers
46 views

Find a linearly independent set of vectors that spans the same subspace of $\mathbb{R}^3$ as that spanned by u, v and w. [on hold]

Consider the vectors $u= (-2, -2, 2)$, $v=(-1, 2, -3)$ and $w=(-6, 0 -2)$. Find a linearly independent set of vectors that spans the same subspace of $\mathbb{R}^3$ as that spanned by $u$, $v$ and ...
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3answers
230 views

What is the main difference between a vector space and a field?

In my opinion both are almost same. However there should be some differenes like any two elements can be multiplied in a field but it is not allowed in vector space as only scalar multiplication is ...
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1answer
33 views

Subspaces and annihilators

I am trying to show this question. My understanding of annihilators is that for a vector space $V$ over $K$, with $S$ being a subset, the annihilator of $S$ is the subspace $S^0$ of linear functions ...
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0answers
17 views

space of solutions of a PDE

so I have just completed part (c) and I'm now on part (d). To fill you in, I have found that $K = -\pi^2 (n^2+m^2)$ for some $n,m \in \mathbb{Z}$ and $p = n\pi, q = m\pi$ now I don't really ...
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0answers
1 views

Taking components of a system containing multiple vectors.

Q. In the arrangement shown in fig. the ends P and Q of an inextensible string move downwards with uniform speed u. Pulleys A and B are fixed. The mass M moves upwards with a speed. My text ...
1
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1answer
41 views

What is the dimension of the vector space of functions $f:\mathbb R\to\mathbb R$?

What is the dimension of the vector space of functions $f:\mathbb R\to\mathbb R$? I want to say that it is at least $2^{\aleph_0}$, but I have no idea how to sharply pin it down otherwise.
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1answer
16 views

Linear functionals and integration verification

Can you please verify my reasoning? (a) Yes as (b) No, as function is squared (c) Yes, same reasoning as (a), squared values of x do not affect linearity. Does the region of integration affect ...
2
votes
2answers
196 views

Sub-dimensional linear subspaces of $\mathbb{R}^{n}$ have measure zero.

I would appreciate it if someone could refer me to a proof (or simply give one here) for the statement in the title. That is: If $k<n$, then every $k-$dimensional subspace of $\mathbb{R}^{n}$ has ...
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0answers
10 views

Where can I find information related to euclidean spaces?

Can you please list some sources where I can study the euclidean space(I am a beginner). Sincerely, I've been trying to understand its meaning and all these symbols, but even the material from the ...
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1answer
12 views

Spans containing line through the origin in direction of vector in the set of the span.

span{u,v} contains the line through the origin in the direction of u. TRUE OR FALSE? The solution manual: "True; the span of u is included in the span of u and v." My answer: FALSE. u and v could ...
6
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1answer
83 views

If $f\in A_k(V)$ and $g\in A_l(V)$ show $i_v(f\wedge g)=i_vf\wedge g+(-1)^kf\wedge i_vg$ - I've got the gist, not sure how to write

If $f\in A_k(V)$ and $g\in A_l(V)$ show $i_v(f\wedge g)=i_vf\wedge g+(-1)^kf\wedge i_vg$ With $A_k(V)$ being the vector space of alternating k-tensors. for $f\in A_k(V)$ for some $v\in V$ we define ...
2
votes
1answer
239 views

Find normal vector of circle in 3D space given circle size and a single perspective

I don't really know what to search up to answer my question. I tried such things as "ellipse matching" and "3d circle orientation" (and others) but I can't really find much. But anyways... I have ...
2
votes
4answers
3k views

How to build a orthogonal basis from a vector?

Anybody know how I can build a orthogonal base using only a vector? I have a vector in the form $v_1 = [a, b, -a, -b]$, where $a$ and $b$ are real numbers. I did try build in the "adhoc way" but, ...
0
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0answers
23 views

Proof involving projections and column spaces

Let $A \in \mathbb{M}_{m×n}(\mathbb{R})$ with linearly independent columns. If $\overrightarrow{b} \in \mathbb{R}^m$, then prove $proj_{Col(A)}(\overrightarrow{b}) = ...
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1answer
40 views

Why is the infinite dimensional vector space with only finitely many nonvanishing components incomplete?

Define a complex vector space $V$ such that any element $\{a_i\}=(a_1,a_2,\dots)\in V$ has only finitely many components $a_i\ne 0$. The inner product is defined as $$(\{a_i\},\{b_j\})=\sum_i^\infty ...
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1answer
31 views

Any isometry is an isomorphism, though the converse is not true. [on hold]

If we define a mapping $f:E \rightarrow F$, where $E$ and $F$ are normed vector spaces, then $f$ is an isometry if $f$ is a linear norm-preserving bijection, that is: $\|f(x)\|=\|x\|, \quad \forall x ...
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1answer
32 views

Why quotient space is needed?

I was wondering why quotient space is so important? Let say for vector space why quotient space is needed? Please explain!
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3answers
38 views

If I have that $L(v)=L(w)$ for all $L \in V^*$, can I conclude that $v=w$?

Let V be a finite dimensional vector space. If I have that $L(v)=L(w)$ for all $L \in V^*$(where $V^*$ is the dual space of V), can I conclude that $v=w$?
3
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1answer
37 views

Canonical isomorphism between vector bundle and dual?

So, we've been asked to show, given a real vector bundle equipped with a metric, that there is a canonical isomorphism from the vector bundle and its dual. Now, there's a theorem that says two vector ...
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0answers
24 views

A counter example of the direct sum of sub spaces.

I was asked to give examples of 3 subspaces where W + V + U is not the direct sum of these 3 subspaces. W, V, and U are subspaces of a vector space, just to clarify. I am having trouble finding out ...
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2answers
17 views

Example of two subspaces $W_1$ and $W_2$ of $ V$ such that $W_1∪W_2$ is also a subspace of $V$

So I know the obvious counter example would be to let: $W_1 = \{(a, 0) | a \in\mathbb{R}\}$ and $W_2 = (0, 0)$. Where $W_1 + W_2 = (a, 0)$ which is an element of $W_1\cup W_2$. But if I wanted ...
0
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2answers
24 views

Find a value of $t$ such that the given vector is parallel to $(2,-3,1)$.

Find a value of $t$ such that the vector $(t^2, -3t, (6-t)^{1/2})$ is parallel to $(2,-3,1)$. I'm pretty sure a vector is parallel if it's a scalar multiple of the other, so I tried setting each ...
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0answers
58 views

My answer to this simple question is wrong but I don't know why

I'm self-studying abstract algebra, and prior to fields there's a brief section on vector spaces. One of the questions asks: "Is $U = \{(a, b-1, c)| a, b, c \in F \}$ a subspace of $F^3$? ($F$ a ...
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0answers
19 views

vector space differential equations

Hi! I am working on some differential equations homework and we are up to the linear algebra part. This particular homework set on Vector space is due, but my teacher has not taught the material yet ...
2
votes
1answer
24 views

Finding dimension of a vector subspace/vectorspace

Let $V$ a vector space with dimension $5$ over a field $\mathbb{F}$ and let $W$ a vector subspace of $V$ with dimension $2$. Define $S=\{T:V\rightarrow V \ |\ T \mbox{ is zero on W}\}$ where $T$ is a ...
1
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3answers
49 views

Find a vector in the matching dimension that is not in the span

I have the following vector $(1,2,-2),(2,-1,1)$. How do I find a vector that is not in the span of those two vectors. I can pick an arbitrary third vector and make the other two vectors equal to it ...
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2answers
552 views

Calculate distance from plane to parallel plane in O using vector and normal

I'm trying to figure out what's the best method to get the distance between two planes where i have the normalized vector of the plane and a point in the plane. What I want to do is to create a ...
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2answers
16 views

Notation for the “scalarization” of a vector with a single non-zero entry

Suppose I have a vector $v$ in the complex space $\mathbb{C}^N$ with only a single non-zero element. Is there a standard notation to replace the vector with a scalar equal to the non-zero value of ...
-1
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1answer
39 views

Linear transformation and characteristic polynomial

Let $V $ be an $n$-dimensional vector space and $T : V \to V$ a non-invertible linear transformation. Show that there is a subspace $W \subset V$ which is $(n−1)$-dimensional and contains ...
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0answers
16 views

Question on the space of all square summable functions involving operator norms and eigenvalues.

Recall that for a set M, $ \mathscr ℓ^2 (M) $ is the space of all square summable functions M $ \to \Bbb C $ . Let $\mathrm T \in Hom( ℓ^2 (\Bbb N) , ℓ^2 (\Bbb N)) $ be given by $$ (\mathrm T a)(n) ...
0
votes
1answer
255 views

Vector space is decomposed into direct sum of its subspace and its orthogonal completement

Lets $E$ be a finite dimensional vector space over the field $k$ and let $g$ be a bilinear form which is symmetric, antisymmetric or hermitian. Let $V$ be a subspace of $E$. Let $V_1 = \{ x| x \in E, ...
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0answers
18 views

Proving that $\left(X(v,v)\right)^{\frac{1}{2}}$ defines a norm on a vector space where X is a positive-definite bilinear form.

Want to show: If $X$ is a positive-definite bilinear form on a vector space $G$ with real-valued scalars and $v\in G$, then $\left(X(v,v)\right)^{\frac{1}{2}}$ defines a norm on $G$. Thus far I have ...
1
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1answer
16 views

Proving the dimension of the intersection of 2 subspaces

Assume that $U$ and $W$ are distinct subspaces $( U ≠ W )$ of a four-dimensional vector space $V$ and $\dim(U) = \dim(W) = 3$. Prove that $\dim ( U ∩ W ) = 2$.
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4answers
36 views

Proof these operations is not a vector space

Let $\mathbb{R}$ be the set of all real numbers. Define scalar multiplication by $\alpha x = \alpha \cdot x$ (the usual multiplication of real numbers) and define addition by $x \oplus y = \max(x, ...
1
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2answers
42 views

Proof that $[v, Tv, T²v, … , T^n v]$ is a basis for $V$ ($dim(V)=n$)?

Let $T:V\to V$ be a linear map from a finite dimensional vector space over a field $F$ to itself. Assume $[v,Tv,T²v,...]$ spans $V$ for some $v \in V$. Don't know at all how to prove that $[v, Tv, ...
1
vote
1answer
33 views

Prove that $z = tx + (1 − t)y$ if $d(x, y)= d(x, z) + d(z, y)$

Let $x,y,z$ be elements of $\mathbb{R}^2$ Prove that $z = tx + (1 − t)y$ if $d(x, y)= d(x, z) + d(z, y)$ d is usual euclidean metric.
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3answers
39 views

If T is diagonalizable then prove that T inverse is diagonalizable. [closed]

If T is an invertible linear operator on a finite dimensional vector space V, then if T is diagonalizable prove that T inverse is also diagonalizable.