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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2
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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
120 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 ...
1
vote
0answers
90 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
70 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
69 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
132 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
884 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
771 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 ...
1
vote
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
93 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
115 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
58 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
149 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
96 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
489 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
249 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
187 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
59 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 ...
1
vote
3answers
458 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 ...
1
vote
0answers
29 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). ...
0
votes
3answers
47 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
150 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
vote
1answer
105 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
votes
1answer
323 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
votes
1answer
33 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 ?
0
votes
2answers
32 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
450 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 ...
8
votes
3answers
180 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 ?
1
vote
1answer
366 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 ...
1
vote
0answers
25 views

Find the vectors such that the parametric curve $g(t)=h_0+ th_1+ t^ 2h_2+ t^3h_3$ passes through points

I'm stuck with this problem: "Find the vectors $h_0...h_3 ∈ ℝ^2$ such that the parametric curve $g(t)=h_0+ th_1+ t^ 2h_2+ t^3h_3$ passes through $(0,0)$ when $t=0$, $(2,1)$ when $t=1$, $(1,3)$ when ...
0
votes
1answer
54 views

Focal point in a parabolic mirror

I'm having trouble with the following problem: "Consider the parabolic mirror given by the equation $z=x^2+y^2$. Show that when the rays of light that travel paralell to the $z$ axis pass through ...
1
vote
2answers
33 views

Finding column and row space without computing A.

I have the a question that asks that I find the column space and row space of: $$A = \begin{bmatrix}1&2 \\4&5 \\2&7\end{bmatrix} \begin{bmatrix}3&0&3 \\1&1&2 ...
1
vote
1answer
42 views

Why do we need “closed” here?

There is a statement such that: Every closed finite co-dimensional subspace of a Banach space is complemented. I don't really see why we need the subspace to be closed. If $X$ is a Banach space and ...
0
votes
2answers
104 views

Finding a basic in subspace in vectors space $\mathbb{R}_3[x]$

In vectors space $\mathbb{R}_3[x]$ we got subspace: $U =$ { p $\in$ $\mathbb{R}_3[x]$; p(1) = p'(1)} and $V =$ {p $\in$ $\mathbb{R}_3[x]$; p(1) = $ \int_0^1 p(t)\,dt. $} How can i find basis of ...
1
vote
1answer
80 views

Are orthogonal spaces exhaustive, i.e. is every vector in either the column space or its orthogonal complement?

Quick question about subspaces, just to make sure I have this straight in my head. Taking an $n\times k$ matrix X with $rank(X)=k$, is every vector in $\mathbb{R}^n$ in either the column space $C(X)$ ...
1
vote
0answers
58 views

Proof by contradiction: $E_1+E_2\doteq E_1 \oplus E_2 \leftrightarrow E_1 \cap E_2=\{0_V\}$

I must proof the following: Prop.: Let $E_1,E_2$ two vector subspace of $V$ then $$E_1+E_2\doteq E_1 \oplus E_2 \leftrightarrow E_1 \cap E_2=\{0_V\}$$ Proof: I must show $$1)E_1+E_2\doteq E_1 \oplus ...
1
vote
2answers
39 views

Show inequality for two elements in $\mathbb{R}^n$

I know that $x,y\in \mathbb{R}^n$ are such that $x_1\leq0,x_1^2\geq x_2^2+\dots+x_n^2$ and $y_1\geq 0,y_1^2\geq y_2^2+\dots+y_n^2$. Is it possible to show that $$x_1y_1+x_2y_2+\dots +x_ny_n\leq 0$$ ...
1
vote
2answers
84 views

About definition of “direct sum of $p$-vector subspaces”

In the books 1 and 2, in Somme directe d'une famille de sous-espaces vectoriels, I am reading the following: 1) let $E,F$ two vector subspaces of $V$, $E+F$ is direct sum, $E+F \doteq E\oplus F$, if ...