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
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 \| ...
0
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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 ...
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votes
1answer
32 views

Line integral of a vector field? [closed]

First of all, sorry for the sketch.. I would be glad if you show me another ways to improve this drawn. I'm studying Line integrals, and this question is really boring me. Please, can someone put ...
4
votes
1answer
36 views

Exterior powers of vector space and kernel

Let $E$ be a vector space and $A$ a subspace of $E$. Let $q$ be a positive integer. Then we can define a subspace $\Lambda^q A$ of the $q$-th exterior power of $E$ by $$ \Lambda^qA=span\{\ ...
2
votes
1answer
34 views

Quotients in a non-discrete valuation ring

Let $R$ be a valuation ring, $\mathfrak{m}$ the maximal ideal of $R$. Let $k$ be the residue field, and $K$ the field of fractions of $R$. Assume that the valuation on $K$ is such that ...
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votes
1answer
61 views

Addition of two subspaces [closed]

Suppose that we have a vector space $V$ and the subspaces $U=\{(x,x,y,y)\in F^4:x,y\in F\}$ and $W=\{(x,x,x,y)\in F^4:x,y\in F\}$. Then how comes $$U+W=\{(x,x,y,z)\in F^4:x,y,z\in F\}?$$
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1answer
17 views

Getting the unique element in the Riesz-Frechet Theorem.

I have this thorem in my book, H', denotes the dual space, that is the set of bounded linear operators from X to the field over X. The way they got the unique element seems very interesting. Does ...
5
votes
8answers
990 views

Is it too much rigor to turn a set into a vector space?

I was reading some online notes on vector spaces and one authors insisted on turning a set $\mathbb{X}$ into a vector space. I thought it was quite insane but maybe I am not seeing the point. The ...
0
votes
1answer
396 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, ...
0
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0answers
36 views

Vector-space of polynomial, find a basis of eigenvectors

Exercise: Let $P_3$ denote a vector space of polynomial of degree 3 or lower (and real coefficients) and let the linear map $F:P_4\to P_4$ be given by $F(p(t))=((t^2+1)p(t))^n$. Find a basis of ...
2
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1answer
26 views

Finding rotated orthogonal vectors without knowing lengths

I have two abstract orthogonal vectors $\mid a\rangle$ and $\mid b\rangle$: $\langle a\mid b\rangle=0$, but I don't know the lengths $\mid a\mid=\sqrt{\langle a\mid a\rangle}$ and $\mid ...
0
votes
1answer
37 views

inner product and hermitian scalar product

suppose $\underline x,\underline y\in\mathbb C^{n\times 1}$ then because the two vectors are in complex vector field, the definition of their inner product will be: $$\langle\underline x,\underline ...
2
votes
4answers
75 views

How to define a vector without a basis?

We know that a vector is an element of a vector space that does not depend by the basis that is chosen to represent it. But, in a finite dimensional vector space, when we want define a specific ...
1
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3answers
84 views

differentiable functions and vector spaces

I am having trouble understanding where to start with the following question: Let $F$ be the set of all differentiable functions on $[a,b]$. Show $F$ is a vector space with the standard operations. ...
1
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1answer
34 views

What functional space does $\mathbb{X} = \{0\}$ belong to?

In a lot of proofs regarding spaces, the example $\mathbb{X} = \{0\}$ is given as the trivial case. Why is that $\mathbb{X} = \{0\}$ is a linear/normed/Banach/Hilbert... space when it is essentially ...
1
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3answers
59 views

Polar Co-ordinate proofs

The expression for acceleration in spherical polars is $$ \ddot{\mathbf r} =( \ddot r -r\dot\theta^2-r\dot\phi^2\sin^2\theta) \mathbf e_r + (r\ddot\theta+2\dot r ...
5
votes
1answer
139 views

Cauchy-Schwarz Inequality without using $\langle a x,y\rangle=a\langle x,y\rangle$

Let $V$ be a vector space and define a function $\langle .,.\rangle:V\times V\to\mathbb{C}$ such that $$\begin{align} & \langle x,y\rangle=\overline{\langle y,x\rangle }\,\,\,\forall x,y\in ...
1
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3answers
40 views

Column Space/Row Space

I just have a small question. I was wondering if someone could explain to me the difference between "column space" and "basis for column space" as well as "row space" and "basis for row space". I've ...
1
vote
1answer
56 views

For linear transformation $f: V\longrightarrow W$. Dim $R(f) + Dim ker(f) = Dim V$.

Just starting linear algebra. For every linear transformation $$f: V \longrightarrow W.$$ dim $R(f) +$ dim $ker(f) = Dim V$ Is this correct? $f(x)=2x$ The range of $f$, $R(f)= R_1$ dimension of ...
0
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1answer
43 views

Is there an error in the solution for this exercise?

I have this exercise: H is a complex hilbert space. And T is a compact operator on H. Show that if H is not separable, then 0 is an eigenvalue of T. Hint: Use lemma 1, and theorem 2. The ...
5
votes
2answers
69 views

What is the implication that $\| \cdot \|_2$ and $\| \cdot \|_\infty$ are equivalent norms on $\mathbb{R^2}$

Given $\mathbb{X}$ = $\mathbb{R^2}$, consider $\| \cdot \|_2$ and $\| \cdot \|_\infty$ We can show that $\| x \|_\infty \leq \| x \|_2 \leq \sqrt2 \| x \|_\infty$ Hence $\| \cdot \|_2$ and $\| ...
4
votes
1answer
152 views

A name for the property $ \| x \star y \| = \| x \| \| y \| $.

Suppose that $ \star: V^2 \to V $ is some binary operation on a vector space $ V $. Should it hold, is there a name for the following property? $$ \forall x,y \in V: \quad \| x \star y \| = \| x \| \| ...
3
votes
2answers
53 views

Finding a basis of a complex vector space over $\Bbb R$ given a basis over $\Bbb C$

Suppose $X$ is a vector space over $\mathbb C$ and has as basis $\{e_1,e_2,\ldots,e_n\}$. Now regard $X$ as a vector space over $\mathbb R$. What will be the basis? My thoughts: I considered ...
0
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1answer
43 views

More on linear algebra vector subspaces

I am continuing on my journey of trying to understand vector subspaces. Question: Let $F(-\infty,\infty)$ be the set of all real-value functions defined at each x in the interval $(-\infty,\infty)$. ...
0
votes
1answer
17 views

Linear transformation of orthogonal complement

Let $T\colon \mathbb{R}^3 \to \mathbb{R}^3$ be linear. If $W$ is a $T$-invariant subspace, then $W^\perp$ (under the standard inner product on $\mathbb{R}^3$) is also $T$-invariant. Is this true? I ...
0
votes
0answers
26 views

Why do uniform, strong and weak convergence coincide for finite dimensional vector spaces?

For linear operators $A_n$, $A$ in a finite dimensional vector space $V$, I am trying to prove the equivalence of $\|A_n - A\| = \sup_{x \in \mathbb{C}^n, |x| = 1} |A_nx-Ax| \to 0$ as $n \to ...
1
vote
1answer
21 views

Cartesian plane points given in equation form

$${x+3\over 5}={y-5\over 3}={z-11\over -3}$$ How would I determine if the above line contains the point $(1, 1, 5)$? Couldn't $(x, y, z)$ be any value? For example $(2, 8, 4)$ than the line would ...
2
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2answers
43 views

Vector in function format

Not sure how to interpret the follow: Find the intersection point(s) of the line $r(t)=(0, -2, -1)+t(1, 1, 1)$ and the plane $x+2y-4z=-3$ Does $r(t)=(0, -2, -1)+t(1, 1, 1)$ mean $r=(0, -2t, -t)$?
2
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4answers
61 views

Vector Subspace

I have a question regarding vector subspaces: show $U=\{A\in M_{22} \mid A^2=A\}$ is not a subspace of $M_{22}$. I have said: let $A={(a_1, a_2, a_3, a_4) = \begin{pmatrix} a_1 & a_2 \\ a_3 & ...
1
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0answers
17 views

How to Construct $N$-dimensional Unitary Matrix Basis

Galitski's Exploring Quantum Mechanics says on its page 29, (There are $N^2$ linearly) independent Hermitian matrices of rank $N$. The number of independent unitary matrices is also $N^2$, since ...
1
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2answers
33 views

Finding orthonormal basis for a subspace $W$ of the Euclidean space $\mathbb{R}^3$.

Problem: Let $\mathbb{R}^3$ be an Euclidean space. Find an orthonormal basis for the subspace $W$ defined as $x + 2y-z = 0$. Attempt at solution: So this is a plane in $\mathbb{R}^3$, so I guess I ...
0
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0answers
31 views

Uncountable Kronecker Delta?

If V and W are vector spaces of uncountably infinite dimension, they still have bases (according to axiom of choice). Let basis sets be $\{v_x\}_{x \in X}$ and $\{w_y\}_{y \in Y}$, and define a set ...
2
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1answer
37 views

Two conflicting answers: Problem in linear algebra involving quotient spaces and T-invariant subspaces

I was presented this scary looking problem in my linear algebra class involving quotient spaces: I am given finite dimensional vector space V over the complex numbers C and linear operator $ T:V ...
0
votes
1answer
31 views

Use Riesz theorem to show functional bounded

I have the linear functional: $ F(v) = \int_\Gamma v \mathbf{g}\cdot\mathbf{n} d\Gamma$ where $\Gamma$ is a (smooth) part of the boundary of a domain $\Omega$, $\mathbf{g}$ is given (assumed smooth) ...
7
votes
4answers
334 views

How can I intuitively interpret this vector operation?

In reading through some very old source code that I inherited and came across a three-dimensional Euclidean vector operation that I can't seem to gain an intuition for. Transcribing the program code ...
0
votes
1answer
38 views

Algebraic subspaces

How do I prove that $U=\{(x,y,z)|x\text{ is an integer}\}$ is not a subspace of $\mathbb{R}^3$? I understand that I have to show $U$ is closed or not closed under vector addition and scalar ...
5
votes
2answers
61 views

$C(M)=\{A\in M_n(\mathbb{C}) \mid AM=MA\}$ is a subspace of dimension at least $n$.

Let $M_n(\mathbb{C})$ denote the vector space over $\mathbb{C}$ of all $n\times n$ complex matrices. Prove that if $M$ is a complex $n\times n$ matrix then $C(M)=\{A\in M_n(\mathbb{C}) \mid ...
2
votes
0answers
28 views

Does $\mathfrak T^r(\Bbb R^m)$ count as an vector space?

Here $\mathfrak T^r (\Bbb R^m)$ denotes all the $r$-th tensors (multi-linear functions) acting upon the elements $(u_1,\cdots,u_r)$ from the product space $\displaystyle \prod^r \Bbb R^m$. And the ...
2
votes
1answer
37 views

Can we state the triangle inequality as $|\int_D f(x) dx| \leq \int_D |f(x)| dx$

$|\int_D f(x) dx| \leq \int_D |f(x)| dx$ is just the infinitestimal version of the triangle inequality commonly presented in any book on vector spaces Can we replace the definition of triangle ...
0
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1answer
95 views

inequality between entries of the vector and $l_2$ norm of the vector

Let $a=(a_1, \ldots, a_n)$ be a vector in $R^n$. I am wondering for which vectors the following would be true: $$ \|a\|_2^2\geq c \sum_{i\ne j,i,j}a_ia_j, \quad i,j=1, \ldots, n. $$ Here $c>0$ is ...
4
votes
2answers
36 views

Parallelogram law in $L_1$ space

Exercise 5.5 from Capinski's and Kopp's book "Measure, Integral and Probability" asks to show that it is impossible to define an inner product on the space $L^1([0,1])$. In order to get this result we ...
2
votes
0answers
64 views

The canonical perspective on the Hodge star operator [closed]

I am looking for the canonical perspective on the Hodge star operator. I want to see it done properly, not using basis for its definition, saying clearly what we assume in its definition. ...
0
votes
0answers
13 views

Projection over product of vectorial spaces

Let $(E = E_1 \times E_2 \times E_n, \|\cdot\|_n)$ be a normed vectorial space product. Can we define the orthogonal projection over $E$ as following: Let $v = (v_1, \ldots, v_n) \in E,$ then $$ ...
2
votes
1answer
21 views

Determining dimension of a sum of subspaces in terms of a parameter

Problem: Consider the linear subspaces \begin{align*} U = \text{span} \left\{ (1,0,1,0), (1,a,0,a)\right\} \quad \text{and} \quad W = \text{span}\left\{(-1, a, a^2, 0), (0,1,0,-1)\right\} \end{align*} ...
1
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1answer
31 views

How do I show that this topology on this linearly-ordered set is regular?

Given some linear ordered set $X$, we define a topology by the basis: all sets of the form $(a,b)$ or $(a,\infty)$ or $(-\infty,b)$, where $a,b \in X$. I need to prove that this topology is regular, ...
4
votes
2answers
59 views

Which $n$-forms are pullbacks of top forms on $\Bbb R^n$

Let $V$ be a finite-dimensional vector space. I write $F_n(V)$ for the $n$th exterior power of the dual vector space. Which elements of $F_n(V)$ can be pulled back from a top form along a linear ...
2
votes
1answer
26 views

Simple question - represent vector with respect to a basis

Basic question here, I've always been weak at this stuff. Suppose that we have a situation $U=WX$ where $U,W,X$ are matrices that are known to us. You can suppose that $U$ is invertible. I want to ...
3
votes
1answer
333 views

Vector Project onto Subspace

So the question is: Let S be the subspace of $\mathbb{R}^3$ spanned by the vectors $ u_2 = \begin{pmatrix} \frac{2}{3}\\\frac{2}{3}\\\frac{1}{3}\end{pmatrix} u_3 = \begin{pmatrix} ...
2
votes
2answers
355 views

If $V$ and $W$ are subspaces of the same dimension such that $V$ meets $W^\perp$, then $W$ meets $V^\perp$

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 ...
-1
votes
2answers
56 views

Prove the Cauchy-Schwarz Inequality (missing a step)

during lecture notes I only caught most of the proof and couldnt write a step down fast enough, and I'm having a touch trouble seeing how to get from the previous step to the next. Here is what i have ...