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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32 views

What are complex and real dimensions of this space?

Show that the vectors $v_1$ = (i,1+i,2+i), $v_2$ = (1,1+i,2+i), and $v_3$ = (2,-i,-i) form a basis for the complex vector space $C^3$.... Show that $v_1,iv_1,v_2,iv_2,v_3,iv_3$ is a basis of ...
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1answer
37 views

Relationship between isomorphic vector spaces and inner product

Let $V$ and $W$ be isomorphic vector spaces. "If $\langle\vec u, \vec v\rangle$ is an inner product in $V$, then it is also an inner product in $W$". Does such a relationship exist?
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0answers
43 views

Algebraic independence of `Riemann-Roch' elements

First of all, I'm not too sure on what terminology should be used in the title: the question deals with the vector spaces $$\mathcal{L(D)}=\{f\colon E\to\mathbb{C} \mid f\text{ is meromorphic}, ...
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1answer
41 views

What do double parallel lines on vectors mean? [closed]

What do the lines mean in the notation $\|u\|$ where $u$ is a vector?
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1answer
53 views

Does $f\otimes_A 1_{A/m}:M\otimes A/m\to N\otimes A/m$ injective for all maximal $m$ imply $f$ is an isomorphism?

Let $A$ be a commutative ring. Suppose $f\colon M\to N$ is a morphism of free $A$-modules of equal, finite rank. If $f\otimes_A 1_{A/m}$ is injective for all maximal ideals of $A$, does this imply ...
2
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1answer
30 views

Basis for a linear space

Let the space of polynomials of $x$ of degree $\leq n-1$ with coefficients in the field $\mathbb{K}$. How can I show that $\{1, x-a, (x-a)^2, \cdots, (x-a)^{n-1}\}$ form a basis for this linear ...
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1answer
15 views

Orthogonal complement of a subspace

I don't understand how to visualize these statements algebraically (are they true or false??): Let $S$ and $T$ be subspaces of $E$: I) $(S+T)^\perp \subset S^\perp \cap T^\perp$ II) $S^\perp + ...
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2answers
40 views

Showing 2 vector spaces are isomorphic.

I am trying to understand how to show two vector spaces are isomorphic. You do this by showing there is an isomorphism that can be mapped between the two spaces. What I don't understand is my ...
1
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1answer
34 views

Is frequency a scalar quantity?

Well, our professor in class posed a question to all of us: Is frequency a scalar quantity? The obvious answer, of course, is yes, it is, because it does not have a direction associated with it. But ...
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0answers
48 views

Let G be an abelian group, and V be a faithful irreducible representation of G over C

This seems kind of obvious to me but I'm really having trouble thinking of what to do! Any help would be appreciated. Let G be a finite abelian group, and V be a faithful irreducible representation ...
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3answers
39 views

Vector spaces $V=-V$

I have a question about vector spaces: Since vector spaces have additive inverses, does that mean that $-V=V$?
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2answers
310 views

Is this proof of Cauchy Schwarz inequality circular or valid?

I'm a college freshman learning linear algebra on my own, and I'm in the section on inner products. I noticed a proof of the Cauchy Schwarz inequality for vectors in my book, and it seems to contain ...
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1answer
16 views

If $\lim\limits_{x\rightarrow 0}(f(x)\cdot y)=0\space\forall y\in\mathbb{R}^n$, show that $\lim\limits_{x\rightarrow 0}f(x)=0$

Let $f(x):\mathbb{R}^n\rightarrow \mathbb{R}^n$ be a function such that $$\lim\limits_{x\rightarrow 0}(f(x)\cdot y)=0\space\forall y\in\mathbb{R}^n$$ Show that $\lim\limits_{x\rightarrow ...
4
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1answer
40 views

Dual Space Isomorphism

If $V$ is a finite dimensional real vector space. Let $$ V^* = \{f: V \to \mathbb{R} : f ~\text{is linear}\} $$ (Note $V^*$ is called the dual space of $V$.) Prove the vector spaces $V$ and $V^*$ ...
2
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2answers
65 views

V is a linear space that has finite dimension $T : V \rightarrow V$ Proof if $Ker T = Ker T^2$ then $Im T = Im T^2$

I have the following question : V is a linear space that has finite dimension $T : V \rightarrow V$ Proof if $Ker T = Ker T^2$ then $Im T = Im T^2$ I tried to proof it, but I got stuck, This is what ...
2
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1answer
34 views

Any tetrahedral geometry theorems of methane bonding angles?

For my 12 grade folio task I need to find alternate ways of finding the bonding angles in a methane molecule (regular tetrahedron). I have already done it through vector methods, co-ordinate geometry ...
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1answer
17 views

volume of the parallepiped spanned by the vectors

Hi I am having difficulty with part (2) of the following proposition. Suppose that $x,y,z\in\mathbb{R}^3$, then (1) $\|x\times y\|=\|x\|\|y\|\sin\theta$ is the area of the parallelogram spanned by ...
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0answers
30 views

Let G be an abelian group. Let V be an irreducible faithful CG-module. Prove that dimV = 1 and G is cyclic.

I was wondering if I could get some help with the following problem. I know how to prove it with Schur's Lemma but I'm having problems without it. Let G be an abelian group. Let V be an irreducible ...
0
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1answer
36 views

Sums of vector space and dimension

In many questions, I see that we have that given $X,Y$ as subspaces of $R^n$ then $V=\{ x+y \mid x \in X, y \in Y\}$ is a subspace. I understand the proof of this. I am not sure why the subtlety that ...
5
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1answer
36 views

Closed formulas for two Poincaré series

Associated with an arbitrary direct sum $E = \bigoplus_{i \ge 0} E_i$, of finite dimensional $k$-vector spaces $E_i$, $i = 0, 1, 2, \dots,$ there is a formal power series $P_E$, with nonnegative ...
9
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2answers
79 views

Exists polynomial satisfying following?

Let $s, u \in M_m(\mathbb{k})$ be a pair of commuting matrices such that $s$ is a diagonal matrix and $u$ is a strictly triangular matrix (with zeros on the diagonal). Put $a = s + u$. Does there ...
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1answer
47 views

What space is the set of all CDFs?

I'm relatively new to functional analysis and am trying to make comparisons across different CDFs (cumulative density functions), i.e. right-continuous, weakly increasing functions ...
5
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4answers
73 views

Not understanding the case where $G$ is abelian with every element of order $2$

Suppose an abelian finite group $G$ (with $o(G)>2$) has every non-identity element of order $2$. Show that there exists a non-trivial automorphism on $G$. After a bit of searching, I found a ...
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1answer
54 views

What is the logic/rationale behind the vector cross product?

I don't think I ever understood the rationale behind this. I get that the dot product $\mathbf{a} \cdot \mathbf{b} =\lVert \mathbf{a}\rVert \cdot\lVert \mathbf{b}\rVert \cos\theta$ is derived from ...
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2answers
29 views

Linear algebra, sub spaces, dimension, span

Consider the subspaces $A=span\{(1,-1,0,0),(0,0,1,-1)\}$ and $B=span\{(1,0,-1,0),(0,1,0,-1)\}$ of $\mathbb{F}^4$. A) what are the dimensions of these subspaces? The dimension of both subspaces is ...
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2answers
21 views

Determine the set is a subspace of vector space

Determine whether the following sets $U$ is a subspace of the indicated vector space $V$. Justify your answer. $$V =\mathbb{R}^3 , U =\{(a,b,c)|a,c \in\mathbb{R}\}.$$ How do we determine and ...
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0answers
23 views

Is there a name for this function with properties…

Let $V$ be a vector space over an algebraic structure $\mathbb{A}$, and suppose we have a binary operation $\star:V^2\to V$. Consider a function $f:V\to \mathbb{A}$ with the property that $$f(x\star ...
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2answers
53 views

The kernel of an element of End(End(V)) for V a finite dimensional vector space

I am a grad student currently studying for an upcoming algebra qualifying exam. I have been working through previous exams and I have gotten through most of them, but I am stuck on this question: ...
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1answer
31 views

Vector Space Spanned by Legendre Polynomials

Problem Let $V$ be the vector space over $\mathbb{R}$ spanned by Legendre Polynomials: ...
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1answer
24 views

Vector norm - Understanding the definition of the unit sphere

If $\|x\|=1$ just means the vector $x$ has length one - Then why is the unit sphere defined as $S=\{x\in X| \quad \|x\|=1\}$? let $X$ be a normed linear space with the Euclidean norm, then letting ...
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1answer
31 views

Vector norm - Understanding it's geometric meaning in regard to the Euclidean norm

I am trying to understand the vector norm. I have a few subquestions to the primary question here, what is the vector norm? 1. Firstly, lets take the Euclidean norm. Is then $\|x\|=d(x,0)$, where ...
3
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2answers
246 views

Examples of infinite dimensional normed vector spaces

In my notes on functional analysis it mentions that $C([0,1]),\ell^p$ and, $\ell^\infty$ are normed vector spaces, and gives some examples of norms that we can define on them. However, it then simply ...
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2answers
40 views

What are some usual norms for matrices?

I am familiar with norms on vectors and functions, but do there exist norms for spaces of matrices i.e. $A$ some $n \times m$ matrix? If so, that would that imply matrices also form some sort of ...
1
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1answer
15 views

Co-ordinate analysis

Coordinates $ (\alpha, \beta, R) $ with $ -1 \ge \alpha,\space \beta \le1,\space R \lt 1 $ are related to Cartesian co-ordinates $ (x, y, z) $ via $ x= R \alpha, \space y= r \beta $ and $ \space ...
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3answers
180 views

Intuitive explanation of div(curlF)=0 [duplicate]

If we consider $\mathbf{F}$ as a vector field, then we say that $\mathrm{div}(\mathrm{curl}(\mathbf{F}))=0$. We can prove this in mathematics easily. But I' am not getting an intuitive explanation due ...
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1answer
40 views

Is it possible to conclude surjective from this theorem?

I have this following theorem in my book : Let $V,W$ be linear spaces over field $F$, let $$B=\{v_1,...,v_n\}$$ is a base of linear space $V$ and let $\{w_1,....w_n\}$ are vectors in linear space ...
2
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1answer
45 views

Is every vector space a banach space?

Using the axiom of choice one can show that for each ($\mathbb{R}$-) vector space $V$ there exists a function $\|\cdot\| : V \rightarrow \mathbb{R}$ so that $(V,\|\cdot\|)$ is a normed vector space. ...
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2answers
30 views

Hahn Banach Theorem: Clarification on meaning of extending a functional?

Hahn Banach Theorem: Given linear (vector) space $\mathbb{X}$, define $u \in \mathbb{L} \subset \mathbb{X}$, $A,B,C$ functionals, A sublinear. $A:\mathbb{L} \to \mathbb{R}, B:\mathbb{L} \to ...
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1answer
25 views

Calculating vector when magnitude changes

If I have a velocity vector $(4, 5, 2)$ its magnitude is $\sqrt{45}=6.7$ m/s Now if this object slows down to $3 \ m/s$ Would the new velocity vector be $(4, 5, 2)\cdot\left({3\over \sqrt45}\right)$ ...
3
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0answers
64 views

Working in a field.

When I calculate vector spaces, diagonalization of matrices, linear transformations on a field, can I work in $\mathbb R$ and ultimately transform the result to that field? For example if I calculate ...
0
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1answer
66 views

Why does a vector space over $\mathbb{C}$, $\mathbb{R}$, or $\mathbb{Q}$ have either 1 element or infinitely many elements? [closed]

Why does a vector space over $\mathbb{C}$, $\mathbb{R}$, or $\mathbb{Q}$ have either 1 element or infinitely many elements? Can anyone help me with the answer to this question?
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2answers
32 views

Linear dependence in vector spaces. Which one is true?

I am trying to determine whether $\{ \sin x, \cos x\}$ is linearly dependent in vector space of all real valued functions. The definition says: A set of vectors $\{ \vec v_1, \dots, \vec v_k \}$ ...
0
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1answer
47 views

Proving if two planes are coincident

How would I prove if the two planes below are coincident or not? $x+2y-5z=1$ $2x-3y+z=3$ Do I need to prove that one equation can equal the other?
2
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1answer
55 views

Clarification on the difference between Brouwer Fixed Point Theorem and Schauder Fixed point theorem

From Zeidler's Applied Functional Analysis Brouwer The continuous operator $A:M \to M$ has a fixed point provided $M$ is compact, convex, nonempty set in a finite dimensional normed space ...
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3answers
27 views

When are norms not equivalent?

There are a lot of questions here on showing that two norms are not equivalent. I understand that two norms may not be equivaelent from their proofs, however I do not understand why this happened in ...
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1answer
41 views

Need help proving $n(T)=n(T^*)$ for finite dimensions.

In my book this is showed: Let H and K be complex Hilbert spaces and let $T\in B(H,K)$. There exists a unique operator $T^* \in B(K,H)$ such that $(Tx,y)=(x,T^*y)$ for all $x\in H$ ...
2
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4answers
58 views

Showing that gravitational flux remains constant.

Let the vector field $$\vec{F}(x,y,z)=\frac{GM}{(x^2+y^2+z^2)^\frac32} \begin{pmatrix} x \\ y\\ z\\ \end{pmatrix}$$ Where $G$ is the universal gravitational constant and $M$ the mass of earth. I ...
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0answers
29 views

. Find the projection of the triangle on the coordinate planes.

Given the following, three vectors: a⃗ =3i−2j+5k b⃗ =i−6j+6k c⃗ =2i+3j−k Relative to cartesian coordinate systems with origin O. I calculated the sides to be 4.58,11.45 and 7.87. I also calculated ...
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1answer
19 views

$| \langle a , i \rangle| \leq \| a\|$ if $\|i\|=1$ this space is a normed vector space upon $\langle , \rangle$ . Why does this apply?

I tried over Cauchy Schwartz to conclude, but could not. Anyone see why this is ? The term: normed vector space upon $\langle , \rangle$ i hear for the first time, Im assuming it means that: $$\|a \| ...
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0answers
30 views

Equivalence of Norms and Open Mapping Theorem

Let $V$ be a vector space with two norms $||\quad||_{1}$, $||\quad||_{2}$, making $V$ a complete normed vector space. Assume $\exists C$ (constant) such that: $||v||_{2} \leq C||v||_{1}, \forall v ...