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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2answers
2k views

Why is cross product only defined in 3 and 7 dimensions? [duplicate]

Why $3$ and $7$? I know from some reading that Hurwitz's Theorem explains this, but can someone help me build some intuition behind this or perhaps provide a simpler explanation? It still seems ...
1
vote
1answer
309 views

How to Prove the Dimension of the Annihilator

I have the same question as posted here . However I don't understand the proof given. The question is, for a Vector Space $V$ with a subspace $U$, prove that: $dim U + dimU^{0}=dimV$. Where $U^{0}$ ...
0
votes
1answer
55 views

Change of basis in linear Algebra

Why do we go for change of Basis ? And where is it used in real time application ? What is the differenc between change of basis and transformation?
2
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2answers
81 views

dimension of a subvector space

Let $V$ be a vector space over field $F$ , $\dim V=n$ and $W$ be a subspace of $V$ , $\dim W=m<n$. We set $E=\{T:V\to V: T ~\text{restrict to}~ W ~\text{is zero}\}$. What is $\dim E$ as a subspace ...
0
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1answer
107 views

Invariant subspaces using matrix of linear operator

I am attempting the following problem but stuck at some parts: How does one find the (2 dimensional) subspaces that are invariant under $A$ for $$A = \begin{pmatrix} 1 & 0 & 0 \\ 0 &2 ...
0
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1answer
77 views

Union of vector subspaces, sum of dimensions of vector subspaces and direct sum of vector subspaces

I am currently reading Linear Algebra Done Right by Sheldon Axler, and I have stumbled upon some proposition that I have trouble verifying. Excerpt from the book: Proposition. Suppose ...
1
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1answer
67 views

About the Fundamental Theorem of Vector Space

In a book I found the following: "In a vector space $V$ of all real valued continuous functions of $x$ defined in the interval $[0,1]$, then $(f+g)(x)=f(x)+g(x),\forall f,g \in V $ ...
0
votes
1answer
236 views

How to prove sum of vectors with same magnitude is equal to zero.

Suppose that we have $n$ vectors $v_1,v_2...v_n$ with same magnitude in plane s.t. the angle between $v_i$ and $v_{i+1}$ is $2\pi/n$ then $v_1+v_2+...v_n=0$ for all $n \geq 2$. I can show this by ...
0
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1answer
95 views

Linear Algebra quick question over isomorphism

can someone provide me an example of two isomorphic subspaces of r2 that are not identical? I am just curious since I can only find ones that are identical
0
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1answer
48 views

Linear Algebra 2 Quick questions regarding my understanding of isomorphism

I know the definition of isomorphism but can you provide me two isomorphic subspaces of $\mathbb R^2$ that are not identical, and an example of a set that spans a subspace of $\mathbb R^3$ but is not ...
1
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3answers
63 views

Linear Algebra Vector Space matrix help

Let $M_{2\times2}$ be a vector space of all $2\times2$ matrices. If the transformation from $M_{2\times2}$ to $M_{2\times2}$ is $t(A)=A+A^T$ and $A$ is a $2\times2$ matrix with the top row $a,b$ and ...
0
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1answer
320 views

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} ...
0
votes
1answer
70 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 ...
0
votes
2answers
33 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 ...
1
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5answers
327 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) = ...
2
votes
1answer
190 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 ...
1
vote
1answer
107 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 ...
0
votes
0answers
39 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 ...
1
vote
0answers
67 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) ...
0
votes
1answer
97 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 ...
2
votes
2answers
51 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
votes
0answers
118 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. ...
6
votes
1answer
133 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
93 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 ...
1
vote
2answers
73 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
votes
0answers
70 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 ...
0
votes
1answer
37 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 ...
1
vote
1answer
18 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 ...
1
vote
1answer
32 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) ...
0
votes
2answers
147 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 ...
0
votes
2answers
954 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 ...
4
votes
1answer
848 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
42 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, ...
0
votes
3answers
78 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
vote
1answer
36 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
votes
3answers
94 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
votes
1answer
119 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 ...
0
votes
1answer
41 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
votes
0answers
31 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 ...
0
votes
2answers
63 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
votes
1answer
68 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
votes
2answers
26 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 ...
0
votes
3answers
150 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
votes
2answers
97 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 ...
0
votes
2answers
50 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)= ...
1
vote
2answers
526 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
votes
2answers
259 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 ...
0
votes
1answer
189 views

Find basis of the annihilator set

$V$ $= \text{span}\{(1,2,3),(1,1,1)\}$ $\subseteq \mathbb{R}^3$. Find the vectors spanning $V^0$ in terms of the usual basis for $(\mathbb{R}^3)^*$. So we want linear functionals $f \in V^*$ such ...
0
votes
0answers
48 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 ...
1
vote
0answers
63 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 ...