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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18 views

Projecting/Mapping 2D vector graphics onto 3D-models (like STL-files)

At my working place, I have the task to map or project 2D-vector graphics onto 3D-models (like stickers). This is not at all about rendering, but I would need the exact 3-dimensional coordinates of ...
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1answer
26 views

determining a linear isomorphism so that two quadratic forms become equivalent

Consider the matrix $$ G = \begin{pmatrix} 3 & 1 & -2 \\ 1 & 2 & 0 \\ -2 & 0 & -3 \\ \end{pmatrix}$$ and the quadratic form $q: \mathbb{R}^3 \to \mathbb{R}$, given by $q(v) ...
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1answer
26 views

Equations of the same plane

Are \begin{equation*} -x-4y+3z=-9~\text{and}~x+4y-3z=6 \end{equation*} equations of the same plane? I graphed them and they look the same, but I am not sure. Thanks
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1answer
37 views

Prob. 14, Sec. 3.8 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: A Hermitian positive semi-definite form

Let $X$ be a complex vector space, and let the map $h \colon X \times X \to \mathbb{C}$ satisfy the following conditions: For all $x, y, z \in X$ and $\alpha \in \mathbb{C}$, (i) $h(x+y, z) = ...
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1answer
22 views

Dual Space of an Euclidian Space is also Euclidic with a specific bilinear form

Let $\gamma: V \times V \to K$ be a nondegenerate bilinear form, and let $\overline{\gamma}$ be defined by: $$\overline{\gamma}: V^* \times V^* \to K, \gamma(x, y) = ...
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3answers
121 views

A question on linear algebra [closed]

Let $V$ be an $n$-dimensional vector space and $T$ be a linear operator on $V$. Condition 1: there exists $0\neq v\in V$ such that $v, Tv,\ldots, T^{n-1}v$ are linearly independent. Condition 2: ...
3
votes
1answer
33 views

Linear combination of matrices over finite and infinite fields

Let $F\subset K$ be the fields. Let $A_1,\ldots, A_m$ be the $n\times n$ matrices over the field $F$, and $c_1,\ldots,c_m\in K$ such that $c_1A_1+\cdots+c_mA_m$ is invertible. How to prove that for ...
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1answer
20 views

What are the least number of extremal points on a polytope?

Given a polytope $P$ embedded in $\mathbb{R}^D$. I can't prove the property that any facet of the polytope will have at least $D$ extremal points lying on it. I can see it for the case $D=2$ but not ...
8
votes
1answer
76 views

Proving that every vector space has a norm.

I am trying to prove that every vector space $X$ has a norm. I have some silly questions, but it's better to ask them now instead of later. I think I'm having a bit of trouble getting intuition about ...
3
votes
2answers
53 views

Subspaces dimensions in $\mathbb{R}^7$

if $U$ and $W$ are subspaces of $\mathbb{R}^7$ and $\dim U = \dim W =4$ then in $U \cap W$ there's a vector different then $0$. I think that it's true, am I correct?
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2answers
19 views

Coordinate vector equation

I have the following bases which are bases of $\mathbb{R}^3$ $$B = ((1,1,1), (0,1,1), (0,0,1))$$ $$C = ((1,2,3), (-1,0,1), (1,0,1))$$ I need to find if this equation is correct $$[(1,2,3)]_B = ...
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0answers
54 views

Change of basis formula proof

So I know that this involves using the chain rule, but is the following attempt at a proof correct. Let $M$ be an $n$-dimensional manifold and let $(U,\phi)$ and $(V,\psi)$ be two overlapping ...
3
votes
1answer
48 views

question about direct sum of vector fields and preservation under quotient spaces

Hello all I was given this question in linear algebra it is two parts and asks to prove or give a counterexample. We are given a vector space V and a subspace of it W and the quotient map $ \pi : V ...
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0answers
26 views

How do I adjust a vector by an amount from another vector

I'm being very dense, but I can't work this out. If I have a direction vector, say (0,1,3) and another vector, maybe ...
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0answers
14 views

Dependence of isomorphism of two vector spaces with Their fields

V and G be two vector spaces over field F and G respectively. Can they be isomorphic if F is not isomorphic to G? What if F is isomorphic to G? I've no idea where to start...Please help
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votes
2answers
73 views

Why is the following set not a vector space?

I have the following set and I want to know whether it's a vector space or not: $W = \{(x, y, z) ∈ \Bbb R^3 : (x + y)(2y − z) = 0\}$ Now, I understand that if I have a set W and it's a vector ...
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2answers
27 views

Need help understanding wiki's informal description of an affine space.

The following was taken from https://en.wikipedia.org/wiki/Affine_space: The following characterization may be easier to understand than the usual formal definition: an affine space is what is left ...
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1answer
30 views

On dual spaces and inner products

Let V be a vector space over $\mathbb{C}$ equipped with an inner product $\langle\, , \rangle:V\times V\mapsto\mathbb{C}$. I need to prove that any linear function $\phi:V\mapsto\mathbb{C}$ (element ...
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3answers
39 views

How to find the basis of the following vector space?

I'm trying to find the basis of the following vector space but I can't seem to be able to find it: $W = \{x = (x_1 , x_2 , x_3 , x_4 , x_5 ) ∈ \Bbb R^5 : x_1 − x_3 − x_4 = 0\}$ I understand that ...
2
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3answers
63 views

Are the coefficients of a vector according to a basis unique?

If I have a vector space $V$ ( of dimension $n$ ) over real numbers such that $\{v_1,v_2...v_n\}$ is the basis for the space ( not orthogonal ). Then I can write any vector $l$ in this space as ...
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0answers
20 views

isometric quadratic spaces over a prime field

Let $(V, \gamma)$ be a quadratic space, where $V$ is an $n$-dimensional $\mathbb{Z}/(7)$-vector space and $r = r(\gamma)$ is the rank of the bilinear form. I want to show: either, $(V, \gamma)$ is ...
0
votes
1answer
28 views

Finding the property of a basis

Let $V = P_2 [x]$, the vector space of polynomials of degree at most 2. Given that $\mathcal B \subset V$, I want to find whether the following is a basis, not linearly independent, not spanning, or ...
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1answer
35 views

Each bilinear form induces a unique bilinear form from the dual space

Let $V$ be a finite dimensional vector space over a field $K$. Let $\gamma: V \times V \to K$ be a nondegenerate bilinear form. I now want to show that there exists one and only one bilinear form ...
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0answers
14 views

finding equality with subspaces direct sum

assume that $U_1 \cap U = \{0\}$ and $U_2 \cap U = \{0\}$ $U_1 \oplus U = U_2 \oplus U$? I thought that it's correct because I could find a counterexample.
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3answers
48 views

How to find the basis of the following vector spaces?

I'm trying, in vain, to find the basis of the following vector spaces: (a) $W = \{x = (x_1 , x_2 , x_3 ) ∈ \Bbb R^3 : x_1 − 2x_2 + x_3 = 0, 2x_1 − 3x_2 + x_3 = 0\}$ (b) $W = \{x = (x_1 , x_2 ...
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1answer
49 views

Number of two dimensional sub spaces of a vector space over a finite field.

Let {$e_1,e_2,e_3,e_4$} br a basis of $4$-dimensional vector space over a finite field with p elements. The number of $2$-dimensional subspaces of $V$ not containing $e_4$ and not contained in ...
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2answers
49 views

Help understanding the range and kernel of a linear transformation

I'm having some trouble understanding the Range and Kernel of a linear transformation. The definition goes as follows: Let $T:V \longrightarrow W$ be a linear transformation. Define the sets ...
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0answers
21 views

Proof of dual norm relation: $\frac{1}{q} + \frac{1}{r} = 1$

Recall: $\|\|$ is a norm in $R^n$, and its dual norm is defined as $\|z\|^*=\text{sup}_{\|x\|\leq1}z^Tx$. If $q$-norm and $r$-norm are dual norm, then we have the following relation: ...
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2answers
38 views

Let $V$ be a $K$ - Vector Space, $f: V \rightarrow V$ a linear map. Under some condition, show that $v, f(v),…,f^n(v)$ is linear independent.

Let $V$ be a $K$ - Vector Space, $f: V \rightarrow V$ a linear map. Let $v \in V$. May a number $n ≥ 0$ exist, so that: $f^n(v) \not= 0$ and $ f^{n+1}(v) = 0$. Show that $v, f(v),...,f^n(v)$ is ...
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1answer
20 views

What is the equation of a facet and it's relation with all points of the polytope?

I was reading about polytopes and I came across about how to define equations of facets defining the polytope. The source I am reading from says If $n$ is a normal vector to a facet $F$ of a ...
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2answers
33 views

Determine interior and boundary of a set

Let $(X,||\cdot||)$ be a normed vector space, where $$X = \big\{ (a_n)_{n \geq 1} ~~|~~ (a_n)_{n \geq 1} \text{ is a bounded real sequence }\big\}$$ and $$\|(a_n)_n\| = \sup_{n \in ...
3
votes
2answers
80 views

Matrix ring $M_2(\mathbb{C})$, $\mathbb{C}^2$ with $M_2(\mathbb{C})$-module structure.

Let $R$ be the matrix ring $M_2(\mathbb{C})$. Let $M = \mathbb{C}^2$ with its natural $R$-module structure (just given by the usual action of $2 \times 2$ matrices on $2$-dimensional vectors). My ...
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5answers
83 views

How to prove there's a vector $z \in \mathbb{R}^4$ orthogonal to two linearly independent vectors $x,y \in \mathbb{R}^4$?

Let $x, y \in \mathbb{R}^4$ with $\{x, y\}$ being linearly independent. Prove that there exists a non-zero vector $z$ that is orthogonal to both $x$ and $y$. Any hints on what to do after the ...
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3answers
39 views

Use Vectors To Show Three Vertices Belong to a Right Triangle

The Full Question Theorems Used This is what I call theorem 1: My Work This problem has two major steps as far as I can see. First, I must show that these are points of a triangle(not ...
2
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1answer
31 views

Understanding the Replacement Theorem (Exchange Theorem)

I'm learning about Basis and Spans and now that's I've figured out what these are, I'm trying to understand the Replacement Theorem(also called the Exchange Theorem). The definition goes like this: ...
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1answer
17 views

Is it possible to find a vector that is orthogonal to this set?

I have a set of four vectors in $\mathbb{R}^4$: $\{ \vec v_1, \vec v_2, \vec v_3, \vec v_4 \}$ The first three are linearly independent, but $ \vec v_4 $ is a linear combination of the others. Is it ...
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2answers
38 views

I'm struggling to find this transformation matrix

$T:\Bbb{P}_3 \to \Bbb{P}_3$ is a linear transformation such that: $$\begin{align} T\left(-2 x^2\right) &= 3 x^2 + 3 x \\ T(0.5 x + 4) &= -2 x^2 - 2 x - 3 \\ T\left(2 x^2 - 1\right) ...
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0answers
19 views

Find a generator for vectorial subspace

S = {$(a, b, c, d) ∈ C^4 : 2ia = b, c + d − ib = 0$} $c+d-i(2ia)=0$ $c+d+2a=0$ $c=-d-2a$ $(a,2ia,-2a-d,d)=a(1,2i,-2,0)+d(0,0,-1,1)$ Is this solution correct?
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1answer
36 views

Cross product of the gradient of two functions

I am having a bit of a confusion with some claims I keep finding on a book of Fluid Dynamics. Let's say we have two functions in 3-D space, $f(\mathbf{x})$ and $g(\mathbf{x})$, with the following ...
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0answers
30 views

Example of the inequality $c_0\neq\bigcup l_p$

As part of an exercise, I was asked to prove or disprove the following proposition: There exists an $x\in c_o$, such that $x\notin l_p$ for every $1\le p\lt\infty$. Before I show my proof, I will ...
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1answer
47 views

Question about conditions for conservative field

Question about conditions for conservative field In common textbooks' discussions about conservative vector field. There is always two assumptions about the region concerned, namely the region is ...
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1answer
36 views

Using SVDs to prove $C(XX^{\prime}) = C(X)$

Let $C$ denote the column space. I would like to prove $C(XX^{\prime}) = C(X)$ for $X \in M_{n \times p}$, $X^{\prime}$ denoting the transpose of $X$. This answer suggests using singular value ...
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1answer
34 views

Find the projection of a vector onto a subspace of $\Bbb R^4$

I need to find the projection of $\vec b = (1,1,1,1)$ onto a subspace of $\Bbb R^4$ described as: $$V=\{(x,y,z,t)\,:\,x=y+t\ \hbox{and}\ 2x=y+z\}\ .$$ Thanks for any help i get guys.
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1answer
16 views

For a matrix $O$ containing columns which are an orthonormal basis for a column space, why does $O^{\prime}O = I$?

Theorem: let $\{o_i\}_{i \in \{1, 2, \dots, r\}}$ be an orthonormal basis for the column space of a matrix $X$ and let $O = \begin{bmatrix}o_1 & o_2 & \cdots & o_r\end{bmatrix}$. Then ...
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0answers
24 views

Why do the vectors perpendicular to [1, 1, 1] and [1, 2, 3] fall on a line, as opposed to a plane?

And what's the intuition here? This is question 6(c) in pset 1.2, Strang's Linear Algebgra, 4th Ed.
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2answers
34 views

Linear operator of infinite dimension

Let $T: V\rightarrow V$ a linear operator with finite dimension. If exists a linear operator $U: V\rightarrow V$ such that $TU=I$, prove that $T$ is invertible. Prove that if the ...
0
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2answers
26 views

Prove statement about projection (linear map)

I am working on the following problem and do am not sure how best to approach it. Let $U$ be a vector space over a field $F$ and $p, q: U \to U$ be linear maps. Assume $p + q = \text{id}_U$ and $pq = ...
2
votes
3answers
42 views

Understanding the difference between Span and Basis

I've been reading a bit around MSE and I've stumbled upon some similar questions as mine. However, most of them do not have a concrete explanation to what I'm looking for. I understand that the Span ...
0
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2answers
45 views

Determine which values of $\lambda \in \mathbb{R}$ cause the following vectors to be a basis

I am working on the following problem. Suppose that $\{v_1, v_2\}$ is a basis of a real vector space $V$. For which values of $\lambda \in \mathbb{R}$ is $\{w_+, w_\lambda\}$ a basis of $V$, where ...
1
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1answer
21 views

What means to complete a pair of vectors $(w, s)$ to an arbitrary basis of $R^d$?

I found in an article this : Let $B = (b_1, b_2, . . . , b_d)$ be an orthonormal basis of $R^d$ such that $<b1, b2 >=< w,x >$ (where $< ... >$ denotes linear span). In order to ...