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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34 views

Checking Axioms is Vector Space

Currently I am studying a section from my book on vector spaces. I'm having issues in understanding how I am supposed to prove some of the questions in the Exercises section, such as: In each of the ...
3
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2answers
31 views

dimension of intersection of a family of subspaces

Let $V$ be a vector space and $\dim V=n$. Let $u_1,u_2,\dots,u_t$ be a family of subspaces of $V$. Assume that if you choose any $n$ subspaces from $\{u_1,\dots,u_t\}$, the dimension of their ...
2
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1answer
36 views

Does $d(x,y)=\sqrt[3]{|x_1-y_1|+|x_2-y_2|}$ define a metric?

$d: \mathbb{R^2} \times \mathbb{R^2} \rightarrow \mathbb{R}$ where $d(x,y)=\sqrt[3]{|x_1-y_1|+|x_2-y_2|}$ for $x=(x_1, x_2)$ and $y=(y_1, y_2)$ I am trying to determine if $d$ defines a metric on ...
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3answers
40 views

Homeomorphism & inverse, between $U=\{ (x,y) \in \mathbb{R^2} :|x|+|y|\leqslant 2 \}$ and $V=\{(x,y) \in \mathbb{R^2} : \max(|x|, |y|)\leqslant 3\}$

Find a homeomorphism, and its inverse, between $U$ and $V$ where: $U= \{ (x,y) \in \mathbb{R}^2 : |x|+|y| \leqslant 2 \}$ $V= \{(x,y) \in \mathbb{R}^2 : \max (|x|, |y|) \leqslant 3 \} $ I have ...
1
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2answers
33 views

Basis of defined subspace

Find a basis for the subspace $W$ of $\mathbb{R}^4 $ , where $$W = \left\{\ \begin{bmatrix}s+4t &\\6s &\\t &\\s\end{bmatrix}: t,s \in \mathbb{R} \right\}$$ I can find basis given ...
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2answers
34 views

Restricting a basis of a vector space $V$ in order to form a basis of a subspace of $V$

We know as a trivial result of Linear Algebra, that given a finite-dimensional vector space $V$ with $\dim V=n$ and $W\subset V$ a subspace of $V$ with $\dim W=k$, $k<n$ we can always extend a ...
3
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1answer
63 views

Deriving Euler-Lagrange Equation

I have just started studying Calculus of Variations, and need some help about deriving the Euler-Lagrange equation. In the book I'm reading, the writer starts by imposing the following inner product ...
0
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1answer
12 views

Free variable located on a pivot position of a matrix?

I'm really confused here. I'm having trouble seeing why is $ x_1 $ a free variable instead of a zero-valued variable, while finding the null space of the following matrix. I shall post my progress: ...
0
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2answers
35 views

Two sets are equivalent if and only if they span the same vector space

The definition they gave me of equivalent sets is: Two sets of vectors {$v_1,...,v_n$} , {$w_1,...,w_n$} are equivalents if every vector of each set can be written as a linear combination of the ...
1
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1answer
31 views

Restriction of an isomorphism to an invariant subspace may fail to be surjective

I'm wondering whether the restriction of a vector space automorphism $f : V \to V$ to an invariant subspace $W \subset V$ can fail to be surjective, i.e. $f\vert_W : W \to W$ is not an ...
1
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1answer
25 views

How to calculate the weights and weight vectors of $Sym^n(V)$

I am wondering how to calculate the weights and weight vectors of $Sym^n(V)$ note: I am working with in $\mathfrak{sl}_2$ From my lecture notes, I know that the weight vectors $Sym^2(V)$ are $v_j$ ...
1
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1answer
27 views

Euclidean norm on integer lattice

Does the Euclidean $ L^2 $ norm (and distance) make any sense on an integer lattice in $ \mathbb{R}^n $? And what is the preferable way of calculating a type of norm in such spaces?
0
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1answer
32 views

Find the extrema of $\sum_{i=1}^n u_i v_i \log \left| \frac{v_i}{u_i} \right|$

This question is similar to the following one: Maximizing and minimizing dot products. However there are significant differences, hence I opened a new question. Maximize and minimize $$\sum_{i=1}^n ...
2
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1answer
52 views

Maximizing and minimizing dot products

Given 2 vectors $u,v \in \mathbb{R^n}$ such that $\|u\| = 1$ and $\sum_{i=1}^n v_i= c$ where $c<1$, I would like to maximize $$\sum_{i=1}^n u_i v_i \log (v_i)$$ and minimize $$\sum_{i=1}^n u_i v_i ...
0
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1answer
29 views

Why for any $(n-l+1)$ orthogonal unit vectors $\{v_1,…v_{n-l+1}\}$ ,$ \sum\limits_{i=1}^{n-l+1}a_{\alpha\beta}v^\alpha_iv^\beta_i\ge0 $

Let $V$ be a n-dimensional vector space,$A=\{a_{\alpha\beta}\} $ is a $n\times n$ antisymmetric matrix ($A^T=-A$), and $rank A=l$. Why for any $(n-l+1)$ orthogonal unit vectors ...
0
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0answers
34 views

Basis of a direct sum

it seems almost trivial to me but I want to double check my proof in case I forgot something. Suppose that the finite dimensional vector space $ W=U\oplus V$ is the direct sum of two veto subspaces ...
0
votes
1answer
21 views

How to find the amount of degrees to rotate a vector to be 90 degrees another vector?

I have a vector V that rotates around an axis K and a vector N all in 3D space. I need to find how much to rotate the vector V around S so that it lies 90 degrees to N. So far I have been doing it ...
2
votes
2answers
46 views

Are all of the following statements equivalent? (vectors and matrices)

I know there's a theorem in linear algebra that has a list of statements that are equivalent, as follows: $\det(A) \ne 0$ $Ax = 0$ has only the trivial solution $Ax = b$ is consistent for every ...
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1answer
22 views

Describe the subspace of $R^3$ spanned by the vectors

Please help me with this question. Describe the subspace of $R^3$ spanned by the vectors in $S = {(−1, 1, 4),(1, −1, 4),(1, 1, 4)}$.
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1answer
29 views

further inspects on vector space [closed]

If you have learned anything about linear algebra, checking whether a space is a vector space is the most fundamental task we must do in the first class. However, it always makes me feel tedious to do ...
0
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1answer
56 views

Set of vectors spanning $\mathbb{R}^k$: does it exist $i\in \{1,…,I-1,I+1,…,n\}$ such that $(x_i-x_I)'\beta=0\Leftrightarrow \beta=0_k$?

Consider a set of vectors of dimension $k\times 1$ $$x_1,..., x_I,...,x_n$$ Suppose that the set of vectors $$x_1-x_I,..., x_{I-1}-x_I, x_{I+1}-x_I,..., x_n-x_I$$ spans $\mathbb{R}^k$. Question: ...
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1answer
29 views

Jacobson radical of polynomial quotient ring.

Let $F$ be a field, and $A=F[x]/(x(x-1)^2)$. 1. Find the ideals of $A$. Which of them are simple or maximal? 2. Find the Jacobson radical, $J(A)$, of $A$. 3. Find two composition series for $A$, as ...
-1
votes
2answers
24 views

Prove that two subspaces of dimension 3 of a vector space of dimension 5 intersect at non-zero points.

We are given a vector space V of dimension 5 We are given that it has two subspaces U & W, both of dimension 3. We are to prove that U intersects W at a vector other than 'zero' My progress ...
1
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0answers
27 views

Let $A:\mathbb R_2[x]\rightarrow \mathbb R_2[x]$ is a linear transformation defined as $(A(p))(x)=p'(x+1)$. Find matrix of $(A(p))(x)$.

Let $A:\mathbb R_2[x]\rightarrow \mathbb R_2[x]$ is a linear transformation defined as $(A(p))(x)=p'(x+1)$ where $\mathbb R_2[x]$ is the space of polynomials of the second order. Find all ...
0
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2answers
27 views

Given two sets: $S_1$ and $S_2$. Check if $S_1$ and $S_2$ span the same subspace of the vector space $\mathbb R^4$.

Given two sets:$S_1=\{(1,-2,1,0),(0,5,-2,1),(2,1,0,1)\},S_2=\{(3,-1,1,1),(1,3,-1,1)\}$. Check if $S_1$ and $S_2$ span the same subspace of the vector space $\mathbb R^4$. Rank of a matrix formed of ...
0
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3answers
36 views

Vectors in XYZ Space with Negative Dot Product?

This question is from a Linear Algebra textbook by Gilbert Strang. I got the answer, but don't understand it complete. Can three vectors in the xy plane have u ⋅ v < 0 and v ⋅ w < 0 and u ⋅ w ...
14
votes
8answers
1k views

Zero vector of a vector space

I know that every vector space needs to contain a zero vector. But all the vector spaces I've seen have the zero vector actually being zero (e.g. $\mathbf{0}=\langle0,0,\ldots,0\rangle$). Can't the ...
2
votes
0answers
36 views

Existence of extension of linear map and existence of subspace complement

Great answer by Asaf Karagila to my question leads me to further questions. Let say we deal with vector spaces over $\mathbb{R}.$ Here are three sentences: For every $V$ and its subspace $E\subset ...
1
vote
1answer
46 views

Existence of vector space complement and axiom of choice

Let say we live in the category of vector spaces over $\mathbb{R}$ or $\mathbb{C}.$ Here are three sentences: Axiom of choice Every vector space has a base. For every vector space $V$ and its ...
1
vote
2answers
59 views

Why is the pseudoscalar called pseudoscalar in Geometric Algebra

It makes sense to call it a pseudoscalar in odd dimensions, because it commutes with all other objects. But in even dimensions it anticommutes, why is it still called pseudoscalar? Further I don't ...
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0answers
8 views

There exists another bilinear symmetric map which is a multiple of $F$

This question is a continuation of the following question. So we have $F: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ a bilinear symmetric map, and $K = \{ v \in \mathbb{R}^n \mid F(v,v) = 0 ...
0
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1answer
31 views

Vector spaces - finding bases in $\mathbb Z_7^2$

Consider the vector space $\mathbb Z_7^2$ = {$(a,b) : a,b$ $\in$ $\mathbb Z_7$}. a) Let $S$ = {$(1,3),(5,1)$}. Prove that $S$ is a linearly dependent subset of $\mathbb Z_7^2$. b) Explaining your ...
1
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2answers
28 views

$F$ is indefinite if and only if $K$ is not a subspace

I get the following linear algebra problem in my class. Let $F: \mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}$ be bilinear and symmetric, and let $K = \{ v \in \mathbb{R}^N \mid F(v,v) = 0 \} $. ...
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0answers
14 views

Tensor product on a hilbert space

If I have the expression: $\langle\phi|\hat{A}$ $ \otimes \hat{I}|\phi\rangle$ $ $ $ $$ $(*) where $\hat{A}$ is a linear operator $\hat{I}$ is the identity operator and $| \phi \rangle ...
1
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2answers
21 views

MVT and twice differentiable function

Assume $f(x)$ is twice differentiable and $|f''(x)|\leq M$ for all $x\in R$. Fix $a \in R$. Use the Mean Value Theorem to show that for any $x \in R, x\neq a$, there is $c \in (x, a)$, such that $$ ...
2
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0answers
178 views

How to list all possible dimension of $\ker{T},\ker{T^2},…,\ker{T^{k-1}}$ and the corresponding canonical forms?

Let $V$ be $5$-dimension vectorspace, and $T:\ V\rightarrow V$ a nilpotent linear transofrmation of order (index) $k$ where $1\le k\le 5$. How to list all possible dimension of ...
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1answer
25 views

Tensor prodct on Hilbert Space

How do I show that $\langle \phi|I \otimes I|\phi\rangle=1$ ? where: $I$ is the identity operator and $\phi \in \mathbb{C^2}\otimes\mathbb{C}^2$
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1answer
50 views

Analysing a subspace

$V$ is an $n-$ dimensional vector space. If $W$ is a subspace such that under any isomorphism from $V$ to $V$ we have $T(W)$ and $W$ have a non trivial intersection.Then what can be said about W?
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1answer
13 views

Evaluate the size of the diagonals of parallelogram constructed on the vectors

$\vec a = 5\vec p + 2 \vec q$, $\vec b = \vec p - 3 \vec q$, $\mid \vec p \mid = 2\sqrt{2}$ $\mid \vec q \mid = 3$ $\angle (\vec p,\vec q) = \frac{\pi }{4}$ Ok, I know that $\vec p \cdot \vec q = ...
2
votes
3answers
83 views

$\int_{A}f^2 = 0$ implies $f(x) = 0 \text{ } \forall x \in A$

Follow-up question to Cauchy-Schwarz for integrals. Here's the basic idea: over the vector space of real-valued square-integrable functions defined on a subset $A \subset \mathbb{R}$, define the ...
1
vote
1answer
59 views

Cauchy-Schwarz for integrals

One of the exercises (3.2) of Izenman's Modern Multivariate Statistical Techniques is for $A \subset \mathbb{R}$, $$\left( \int_{A}fg\right)^2 \leq \left(\int_{A}f^2\right) \left(\int_{A}g^2\right)$$ ...
0
votes
1answer
38 views

Sufficient (or sufficient and necessary) conditions for a set of vectors to span $\mathbb{R}^k$

Consider a set of vectors $k\times 1$ $$x_1,x_2,...,x_n$$ I want to check whether this set of vectors spans $\mathbb{R}^k$. How can I do it? Are there sufficient (or sufficient and necessary) ...
0
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1answer
47 views

Tensor product of operators

How do I show that: $(\hat{\sigma} $ $\otimes $ $\langle \Phi|\hat{I} )(\hat{\sigma} $ $\otimes $ $ \hat{I})|\Phi\rangle$ $ =1 $ (The parenthesis many not be strictly correct here) ...
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1answer
28 views

Representation of a line as the span of two vectors

According to Harley, Zisserman (Multiple View Geometry) a line can be represented via the span of a matrix $W^T$ defined as: $$ \begin{bmatrix} A^T\\ B^T \end{bmatrix} $$ where $A$ and $B$ are two ...
0
votes
1answer
24 views

Using the definition of the operator norm

I am given the following problem: Using the definition $$\lVert L \rVert_{\text{op}}=\sup_{\vec{u} \in \mathbb{R}^d, \lVert \vec{u} \rVert=1}\lVert L\vec{u} \rVert$$ of the operator norm of a ...
0
votes
0answers
15 views

Choose a Daubechies wavelet to approximate a polynomial function

Assume I have a polynomial function $s(x) = \sum\limits_{i=0}^d a_i x^i$, and $\boldsymbol{s}_{d}$ is a discretized sample from $s(x)$, i.e. $\boldsymbol{s}_{d}=(s(0), s(\Delta x), \ldots, ...
0
votes
1answer
53 views

How to prove or disprove matrix $A$ is invertible $\iff$ $\det{A}\ne 0$ if it's defined over field $F_p$

How to prove or disprove matrix $A$ is invertible $\iff$ $\det{A}\ne 0$ if it's defined over field $F_p$ What's special about $A$ if it's defined over field $F_p$? For normal procedure, we have $A$ ...
0
votes
0answers
37 views

How to show $T: C^n\rightarrow C^n$ defined by $T(x)=\bar{x}$ is not a linear tranformation?

We write $x=(x_1,\ldots,x_n)$, then $T(x)=(\bar{x}_1, \bar{x}_2,\ldots,\bar{x}_n)$ $T(ax)=(\overline{ax}_1, \overline{ax}_2,\ldots,\overline{ax}_n)=$ Here I am not sure what to write as I don't know ...
0
votes
1answer
27 views

How to verify $T(f)=df/dt(=du/dt\ +\ i\ dv/dt)$ for $T:\ V\rightarrow V$ is a complex linear transformation?

Let $V$ be the vector space of differentiable functions $f:{R}\rightarrow C$. Define $T:\ V\rightarrow V$ by $T(f)=df/dt(=du/dt\ +\ i\ dv/dt)$. How to prove it's complex linear transformation? ...
0
votes
2answers
26 views

How to prove $\alpha =\{1, x,…,x^n\}$ is a basis for $P_n(F)$

Let $v=a_0+a_1x+...+a_nx^n$, if $a_0+a_1x+...+a_nx^n=0$, then all coefficient $a_i\in F$ are $0$ as variables are all with different degrees so it is linearly independent (Here I am no sure how to ...