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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Linear Algebra Vector True and False Questions

I have a few true and false questions. I have explanations for them could someone please check them over? $R^3$ contains two disjoint subspaces. I think this is true for example {1,2,3} and {4,5,6} ...
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23 views

How can I get a rotation angle from a 2d vector?

I have a 2d vector (x,y). And I'd like to obtain from it a rotation angle. For example: I would have 0° degree when (x = positive, y = 0), more than 0° degree when (x = positive, y = positive), and ...
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2answers
25 views

how to impose binarity constraint in a vector

This is part of a homework problem. In an optimization problem, I need to have a K dimensional vector S, such that each entry of the vector is either 0 or 1, and $l_1$ norm of S is <= K. I can't ...
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26 views

Find basis so Transformation Matrix will be diagonal

$e_1,e_2$ will be basis for $V$. $W$ has a basis $\{e_1+ ae_2,2e_1+be_2\}$. Choose an $a,b$ s.t. that the basis for $W$ will have a transformation matrix $T$ will be in diagonal form. $T(e_1) = ...
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1answer
41 views

Understanding the significance of row space and column space basis

I've just learned about the row and column space basis and I'm confused about what the significance of each is. My professor basically hasn't said much and has danced around any direct questions on ...
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1answer
49 views

Intuition for the fact that, in a vector space V over a field F, av = 0 $\implies$ a = 0 or v = 0. (a $\in$ F, v $\in$ V).

I have no trouble proving this: Let av = 0. If a = 0 then then we are done. Otherwise, there exists $a^{-1} \in F$ such that $a{^-1} a = 1$. Multiplying both sides of the equation by $a^{-1}$ gives ...
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28 views

How to define conditions under which linear maps are injective?

In this book (http://linear.axler.net/) proposition 3.2 states the following: Proposition 3.2: A linear map $T : V \rightarrow W$ from vector space $V$ to vector space $W$ is injective if and only if ...
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0answers
32 views

Diagonalization of a linear transformation in the polynomial vector space

Let $V = R_3[X]$ be the vector space of polynomials with real coefficients of degree at most 3 and consider the linear transformation $V \rightarrow V$ defined by $f_a(p(x))=p(1-ax)$ for each $p(x) ...
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1answer
31 views

Polyhedron's Representations and spanning the Euclidian space

Let's say you have to different representations of the same polyhedron $P\neq \emptyset$: $$P=\{x\in \mathbb{R}^n\;|\;h_i^Tx\leq c_i, i=1,...,k \} =\{x\in \mathbb{R}^n\;|\;g_j^Tx\leq d_i, j=1,...,l ...
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2answers
28 views

Moving segments colliding

I need to check if the edges of two triangles will collide. I do an edge-edge check for each pair. One segment is stationary, while the other moves with constant velocity. The segments are not ...
2
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0answers
48 views

Real and complex vector spaces

Suppose that $V$ is a real finite-dimensional vector space and let $V_\mathbb{C}=V\otimes_{\mathbb{R}}\mathbb{C}$ be its complexification. Now let $W\subset V_\mathbb{C}$ be a complex subspace. ...
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1answer
42 views

Duality mappings on finite-dimensional spaces

I have a few questions regarding some concepts in a book "nonlinear partial differential equations by Roubicek" I am studying. The following is from the text. "Let $V$ be a separable, reflexive ...
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0answers
47 views

Is my understanding of an annihilator correct?

This is how I understand the annihilator now, but I feel like it might be incorrect. So for some $U \subset V$, the annihilator of $U$ is all of the linear functionals $t(v)$ in $V'$, such that ...
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2answers
31 views

Cauchy-Schwarz in complex case, using discriminant

There is a proof of the real case of Cauchy-Schwarz inequality that expands $\|\lambda v - w\|^2 \geq 0 $, gets a quadratic in $\lambda$, and takes the discriminant to get the Cauchy-Schwarz ...
2
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0answers
66 views

Isomorphism,on ${R}^4$

I dont understand what the function is for part (a) such that a mapping from $X\in T_p{R}^4$ to $w(X,-)\in T^{\star}_p{R}^4$ be an isomorfism!. So Consider on ${R}^4=(x_1,y_1,x_2,y_2)$ the ...
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1answer
27 views

Not subspace of vector space

I am working on some example of vector subspace. I have this: $\mathbb{R}^2 := \{a\in \mathbb{R}³ ; a = a_1*e^1 + a_2*e^2 + 0*e^3 \}\\$ And I want to prove that ...
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0answers
33 views

Equivalent of SVD for 3D motion capture time series

I work in image science but would like to analyse some motion capture time series data. If it were an image time series, where each point has a scalar value, I would run an SVD to explore the ...
1
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1answer
16 views

Linear combination of solns of differntial solns, any geometric explanation?

Just learned that if $y_1$ and $y_2$ are solutions to a homogeneous equation, then so a linear combination of $y_1$ and $y_2$. Now, I am sure, but don't know enough if there is some geometric ...
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1answer
24 views

What is fixed in a equation in a polynomial vector space

From what I've learned, an equation $p(t)$ in $P_n$ is defined $$p(t) = a_0+a_1t+a_2t^2+\cdots+a_nt^n \tag 1$$ Given the basis $\beta=\{1,t,t^2,\ldots,t^n\}$, $p(t)$ can be written in the form $$p(t) ...
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2answers
25 views

inner product space definition

I have some problem in the definition of inner product space. The book I use to learn in linear algebra and its application 4th edition (David C.Lay) In the chapter 6.7 it define the inner product ...
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1answer
46 views

Find tangent vector to surface given a point on the surface and its normal vector (for a sphere)

I need to know how to find a tangent vector to a point on the surface of a sphere if I am given the point P and the normal vector at that point N. I know that there are many possible tangent vectors ...
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1answer
48 views

Dimension of the vector space of homogeneous polynomials

Let $k[X_0, X_1, \ldots, X_n]_d$, or briefly $k[X]_d$, be the $k$-vector space whose elements are the zero polynomial and homogeneous polynomials of degree $d\geq 1$. I found the following formula for ...
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2answers
39 views

Problem with vector multiplication

I have this plane problem and the answers are released for it. I don't understand this specific part: Why does : (i + 4k) x (3j - k) = -12i + j + 3k. I tried using the cross product method, however, ...
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3answers
64 views

Proof: $V$ and $W$ Vector Spaces, with finite $\dim (n)\ge1$ and $\gamma:V \to W$ an isomorphism

Proof: $V$ and $W$ Vector Spaces, with finite $\dim (n)\ge1$ and $\gamma:V \to W$ an isomorphism, prove that $(\alpha_1,\dots,\alpha_n)$ is base of $V$ if and only if $(\alpha_1,...,\alpha_n)$ is ...
1
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1answer
31 views

to show $\sum_{i=1}^{\infty} |x_i y_i|$ converges

$X$ consists of sets of the form $(x_1, x_2, x_3, \dots)$ where $x_i \in \mathbb R$. Suppose $\sum_{i=1}^{\infty} x_i ^2$ converges. Show that : $\sum_{i=1}^{\infty} |x_i y_i$| converges. where $x,y ...
3
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4answers
78 views

Prove that for every vector $V$, $||V||_{\infty} \leq ||V||_2 \leq || V||_1$

$\newcommand{\inf}{||V||_\infty}$ $\newcommand{\two}{||V||_2}$ $\newcommand{\one}{||V||_1}$ Prove that for every vector $V$, $\inf \leq \two \leq \one$ I have tried to look online for a solution to ...
2
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1answer
48 views

Prove that $DT = I_v$, $TD \neq I_v$, where $D$ = differentiation operator and $T$ is integration

Let $V$ be the linear space of all real polys $p(x)$. Let $D$ denote the differentiation operator, and let $T$ the integration operator that maps each polynomial $p$ onto the polynomial $q$ given by ...
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1answer
39 views

A basis of this vector space?

I am looking for a basis of the set of solutions of $u_{n+2}=u_{n+1}+u_{n}$... Is there some easy basis? I know that all solutions are determined by $u_0, u_1$ but I don't know how to find a basis. ...
2
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0answers
28 views

What is the most generic algebraic structure for which we can define a tensor product? [duplicate]

We can define a tensor product of two vector spaces. But vector spaces are themselves modules and we can also define a tensor product of two modules. My question is the following: are modules the ...
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2answers
36 views

Is every tensor an element of a vector space?

As, the tensor product of two vector spaces $V$ and $W$ over a field $K$ is another vector space over $K$, is it true to say that every tensor is an element of a vector space ? (if we do not consider ...
0
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1answer
27 views

displacement between vectors in 3D

I have a problem with a homework question. The question reads as follows: A particle starts from a position R1 = <2.9, 4.5, 3.3> m. It is then moved through a displacement of $\delta$R2 = ...
3
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2answers
23 views

Proving $L(S)= \cap_{S ⊆ W}\space W$

Let , $S$ be a subset of a vector space , then how do we prove that $L(S)$ , the linear span of $S$ , is the intersection of all subspaces containing $S$ i.e. $L(S)= \cap_{S ⊆ W} \space W$ ? ( I ...
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3answers
57 views

Explain Normalization in Layman's term

Can someone explain me what is Normalization in Layman's term ? If we have a vector a, we normalize it by dividing it by |a|. That is $$\frac {a}{|a|} $$ Why we need normalization?
2
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2answers
56 views

Using a non-zero wedge product to write a set of vectors as a linear combination of another set of vectors in a finite dimensional space.

Question: Let $V$ be a finite dimensional vector space, and let $ \{ v_1, ..., v_r\}$ and $\{w_1, ..., w_r\}$ be two sets of vectors in $V$. Suppose that $\sum_{i=1}^{r} v_i \wedge w_i = 0$, and ...
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2answers
40 views

Show that the functions are vectors.

Let $V$ be the subspace of $C^1(\mathbb R)$ spanned by $f(x) = \sin x $ and $g(x) = \cos x$. a) Show that for any constant value of $\theta$, the functions $f_1(x)=\sin (x+ \theta) $ and $f_2(x)= ...
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2answers
89 views

Find bases for subspaces spanned by vectors.

The standard basis for $P_2(\mathbb R)$, the vector space of quadratic polynomials of the form $ax^2+bx+c$ is the set $S=\{1,x,x^2\}$. Find bases for the subspaces of $P_2(\mathbb R)$ spanned by the ...
3
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2answers
75 views

What is (fundamentally) a coordinate system ?

Consider the following construction of vectors and points. Let's start with a vector space, or more specifically a coordinate space $F^N$ over a field $F$ and of $N$ dimensions. The elements of this ...
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0answers
23 views

Understanding 2nd half rank-nullity theorem proof.

I'm trying to understand the second half of the rank-nullity theorem (the part that shows $T(e_{k+1}) \dots T(e_{k+r})$ is independent). Assume $e_1 ,\dots e_k, e_{k+1}, \dots e_{k+r}$,is a basis for ...
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0answers
34 views

Computationnal geometry: vector, basis, point and coordinate system?

I am trying to build a small geometrical library in C++, that is mathematically consistent (not so false). The goal here is to construct two concepts: vectors and points. I am not sure that the ...
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3answers
58 views

Origin in vector space?

In the wikipedia article about vector space I do not understand this sentence Roughly, affine spaces are vector spaces whose origin is not specified. A vector space does not need an origin. When ...
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0answers
24 views

Help modeling 3d vector field

I'd like some help in finding the correct mathematical description of the stuff below: A sheet is deformed by a mass on it (like in one of those pictures showing the effects of General Relativity). ...
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3answers
42 views

A question about eigenvectors.

Let $T\in L(V,V)$, and let $\{v_1,v_2,\dots,v_n\}$ be a basis of $V$ consisting of eigenvectors of $T$, belonging to eigenvalues $a_1,a_2,\dots,a_n$ respectively. Then $Tv_i=a_iv_i$. Prove that ...
3
votes
1answer
78 views

Can you make the circle into a vector space?

I thought maybe use a set with elements of the form $e^{i\theta}$ but what field would the scalars be taken from? Multiplying $ke^{i\theta}$ with $k \in \mathbb{R}$ or $\mathbb{C}$ doesn't give you ...
1
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1answer
46 views

Linear Algebra, meaning of 0 determinant in linear transformations

Lets say the area of a figure in $\Bbb R^2$ was $10$. Then after a noninvertible linear transformation from $\Bbb R^2$ to $\Bbb R^2$, is there enough info to determine the new area? Since its ...
5
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0answers
200 views

Span and Dimension: A subspace

If $A$ is finite set of linearly independent vectors then the dimension of the subspace spanned by $A$ is equal to the number of vectors in $A$. This is obviously true. Since $A$ is a finite set of ...
0
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1answer
30 views

normal to hyperplane $ 0.5x_1-5.5x_2-2.5x_3+9x_4 \le 0 $

I want to find the equation of the the normal to the hyperplane $$0.5x_1-5.5x_2-2.5x_3+9x_4 \le 0.$$ How can I find that ?
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2answers
19 views

Extending Continuous Basis

It is given $(k-d)$ continuous vector-valued functions $K_1,\dots,K_{k-d}:\mathbb{R}\mapsto\mathbb{R}^k$, with $d\leq k$. Suppose that for all $x\in\mathbb{R}^k$, the set ${\cal ...
0
votes
2answers
45 views

Find a basis for s prep

Q. S is a subspace of R^3 containing only the zero vector. If S is spanned by (1,1,1) and (1,1,-1) what is a basis for S perp? This is what I have so far -> a+b+c = 0 and a+b-c = 0. 2a+2b = 0 ...
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3answers
111 views

A question on vector space over an infinite field [duplicate]

Can a vector space over an infinite field be a finite union of proper subspaces ?
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1answer
48 views

Rigorous definition and relations between point/vector/affine space/vector space/basis/frame/coordinate system

I am trying to understand the exact relation between all these things: point vector affine space vector space basis frame coordinate system Can you explain me rigorously (in the mathematical ...