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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1answer
15 views

Calculating quadrant facing from a rotational matrix and two 3d vectors

I am working on a space-ship simulator, and having trouble with facing arcs between two space objects. Each object has a rotation matrix defined as follows: ...
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1answer
35 views

Question about dimension of a subspace

Let $K$ be a field and define the following subspaces $$V=\textrm{span}(e_1,e_2,e_3),\;\; V^\bot = \textrm{span}(e_4,e_5,e_6)$$ inside $K^6$. Let $\dim L=4$ and assume that $\dim L\cap V\leq 1$. Can ...
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1answer
55 views

The vector space $L(X,Y)$ of linear maps.

Here's a definition on : The vector space $L(X,Y)$ of linear maps. Let $L(X,Y)$ be the set of all linear functions $T:X\rightarrow Y$ .Then $L(X,Y)$ is itself a vector space. The linear ...
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1answer
13 views

How to denote that vector must have one non-zero entry.

How to denote a vector of integers that contains one and only one non-zero entries.
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1answer
12 views

Intersection point between a line and plane: what's wrong with my calculation?

I'm trying to calculate the intersection point between a line and a plane, but apparently there is something wrong with my calculation and I don't know what exactly. The exercise goes as follows: ...
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1answer
15 views

Intersection point between a line and a plane?

So we have a line, let's called it line L, that passes through (2,−2,1) and (−4,1,−3). We also have a plane, let's call it V, that is given by the equation 3x + 4y + 4z = -42. How can I now ...
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0answers
40 views

Basis of $\mathbf{Q}[x]$

I wanna show that the binomials $\binom{x}{k}$ for $k=0,1,\ldots$ form a basis of the $\mathbf{Q}$-vector space $V=\mathbf{Q}[x]$. I can show that for fixed $m\in\mathbf{N}$ the $\binom{x}{k}$ ...
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0answers
13 views

Use of Matlab to put equation into vector form

Is there a way to put the following equation of a line into vector form using Matlab? $\displaystyle y=\frac{cos(s_n)-cos(s_{n+1})}{sin(s_{n+1}-sin(s_n)}(x-sin(s_n))-cos(s_n)$
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1answer
22 views

Deduction of vector form of Snell's law

I was unable to find the deduction of the vector form of Snells's law. $$n_1\sin\theta_1 = n_2\sin\theta_2$$ Here is the vector form, from the article A Theory of Multi-Layer Flat Refractve Geometry ...
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0answers
24 views

linear algebra question

Consider $n$ convex polytopes $S_1, \cdots, S_n$ and a set of matrices $W$ such that each matrix $A\in W$, we have that the $i$-th row of $A$ is a member of $S_i$. (In general $W$ is infinite.) ...
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1answer
30 views

How can you define vectors with complex numbers?

For real vector space, you can define vectors of $x$, $y$, or $z$ on Euclidean space $$x=\left(x_1, x_2, x_3, \dots, x_n\right) \qquad y=\left(y_1, y_2, y_3, \dots, y_n\right)$$ For example, in ...
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1answer
34 views

$dim_\mathbb C V=n$ then $dim _\mathbb R V=2n$

Prove that if the dimension of a vector space $V$ over $\mathbb C$ is $n$ then the dimension of $V$ over $R$ is $2n$ I wanted to do it using isomorphisms i.e. every finite dimensional vector space ...
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0answers
16 views

Vectors and Projection

So i have 3 points. E = (0, 0 ,-5) C = ( 0, 0, 0) S = (-5, 0, 9). I am given several equations that work perfectly. I understand how to do the math, but I am trying to visualize whats going on. So ...
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0answers
17 views

How to prove for an operator $L$ on a vector space $V$ that $Null(L^k)\subset$ $Null(L^{k+1})$?

This was a past exercise and I still struggle to understand why it is necessary to prove it. I could very well be doing it wrong too! We have $L$, an operator (I'm assuming linear, but feel free to ...
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1answer
17 views

A Quotient space Problem

Prove that there is a natural isomorphism between $(V/W)'$ and $W^0$ where $W^0$ is the annihilator of $W$ and $(V/W)'$ is the dual of $V/W$
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2answers
41 views

Prove that $L(V,W)$ forms a vector space

Let $V$ and $W$ be vector spaces over a field $F$. Let $L(V,W) = \{T:V\to W : T \text{ is linear} \}$, that is, $L(V,W)$ is the collection of all linear functions from $V$ to $W$. For $S,T \in L(V,W)$ ...
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0answers
24 views

Show that the orthogonal projection in vector space. [on hold]

Let V be a vector space and U \subset V be a finite dimensional sub-space. Show that the orthogonal projection P_{U}: V\rightarrow U is clearly.
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23 views

Calculating area

I have a triangle defined by its vectors. The triangle itself is intersected by a plane (z=0). My probpem is: I want to calculate the area of the triangle above z=0 and below z=0. I hope you ...
2
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1answer
24 views

Hamel Basis Exercise Proof Clarification.

While looking up something else on stack exchange, I ran across this question An exercise about a Hamel basis and it intrigued me. The answer was provided by Jonathan Golan ...
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0answers
38 views

Linear Algebra (Basis)

We have B: Question: Find a basis in $\mathbb{M}_{3,2}(\mathbb{R}$) that has B. Obs.: I have no idea how to do this. I know that a combination of a basis is a vector in the subspace formed for this ...
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0answers
17 views

Linear algebra (Coordinates)

Question: Find the coordinates of $x=(1,0,0)$ in relation to base $$B=\{(1,1,1),(-1,1,0),(1,0,-1)\}.$$ I tried: $a,b,c\in R$ such that $$a(1,1,1)+b(-1,1,0)+c(1,0,-1)=(1,0,0)=x$$ but I'm not sure ...
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2answers
180 views

Is this set neccesarily to be a vector space?

Suppose $F$ be a field and $S$ be a non empty set such that 1) $a+b \in S $ 2) $ \alpha a \in S$ for all $a,b \in S$ and $ \alpha \in F.$ Is always $S$ to be a Vector space?
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1answer
61 views

What is needed to make Euclidean spaces isomorphic as groups?

Consider the abelian groups $G_n=(\mathbb R^n,+)$ for $n\geq1$. Claim: For any $n$ and $m$ the groups $G_n$ and $G_m$ are isomorphic. This claim is true if one assumes the axiom of choice, and I ...
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1answer
40 views

Exercise 1.1.3 in Charles Weibel’s book “An Introduction to Homological Algebra”

I am trying to teach myself some homological algebra and I got stuck right at the start with Exercise 1.1.3 from the book “An Introduction to Homological Algebra” by Charles Weibel. Exercise 1.1.3 ...
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1answer
21 views

Is this an Alternative Proof of a set of vectors forming a basis?

This is one of my exam past paper question So I proved this correctly by following the normal method which is showing that a, b and c are linearly independant My proof - When I looked at the ...
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1answer
30 views

Find a basis of $M_2(F)$ so that every member of the basis is idempotent

Let $V=M_{2\times 2}(F)$ (the space of 2x2 matrices with coefficients in a field $F$). Find a basis $\{A_1,A_2,A_3,A_4\}$ of $V$ so that $A_j^2=A_j$ for all $j$. My attempt. Let $A_j$ be ...
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1answer
36 views

Nonhomogeneous Linear Systems and Vector Space Solutions

Are there any nonhomogeneous linear systems with a solution set that forms a vector space? I know that, in order to be a vector space, a set must consists of a set V together with operations + (called ...
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1answer
33 views

being $\mathbf{w}$ a vector, how do I calculate the derivative of $\mathbf{w}^T\mathbf{w}$?

Let's say that I have a vector $\mathbf{w}$. How can I calculate the derivative in the following expression? $\frac{\mathrm{d}}{\mathrm{d}\mathbf{w}}\mathbf{w}^T\mathbf{w}$
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25 views

being $\mathbf{a}$ and $\mathbf{b}$ two vectors with same length, how do I expand $(\mathbf{a}^T\mathbf{b})^2$?

Let's say that I have two vectors $\mathbf{a}$ and $\mathbf{b}$. Assuming that they have same length, their product $\mathbf{a}^T\mathbf{b}$ and its square $(\mathbf{a}^T\mathbf{b})^2$ are scalars. ...
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0answers
31 views

Is a vector space over $\mathbb{C}$ also a vector space over $\mathbb{R}$?

Let $V = \{(a_1, a_2,\ldots, a_n):a_i$ is an element of $\mathbb{C}$ for $i = 1,2,\ldots, n\}$; so $V$ is a vector space over $\mathbb{C}$. Is $V$ a vector space over the field of real numbers with ...
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0answers
19 views

Find real solution for an inhomogene system

I have an inhomogene differential equation system $\begin{pmatrix}\dot{x}_1 \\ \dot{x}_2\end{pmatrix} = \begin{pmatrix}-1 & 3 \\ -3 & -1\end{pmatrix} \begin{pmatrix}x_1 \\ x_2\end{pmatrix} + ...
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1answer
24 views

Proof linear independency lemma

If $\mathbf{u}$ and $\mathbf{v}$ is in the complex vector space $V$ and $\mathbf{w}_1 = \mathbf{u} + i \mathbf{v}$ and $\mathbf{w}_2 = \mathbf{u} - i \mathbf{v}$ are linear independent then will the ...
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1answer
29 views

Transformation from cartesian to polar Coordinates of Vector Field

This is fairly low-level, still I would like to get a verification: I vector field $$\mathbf{F}=F_x \hat{e_x} + F_y \hat{e_y} = F_r \hat{e_r} + F_{\phi} \hat{e_\phi}$$ given in cartesian coordinates, ...
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1answer
16 views

End(V) and End(V)xEnd(V) are isomorphic

Let R=End(V) be the ring of all linear endomorphisms of an infinite dimension complex vector space V with countable basis $\{e_{1},e_{2},...\}$ . Prove that R and RxR are isomorphic as left R-modules. ...
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1answer
15 views

Qtuotient space and affine space

Sorry for many questions in this part. But I am still confused about the following: From textbook "Optimization by vector space"(Luenberger): Problem: I read the def. of quotient space many ...
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0answers
29 views

Relation between basis elements under automorphisms

Let $e_1, e_2, e_3$ denote the standard basis for the vector space $\mathbb{R}^3$, and let $f, g: \mathbb{R}^3\to \mathbb{R}^3$ be linear maps such that $g\circ f = f\circ g = {\rm id}$. Also, let ...
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+50

Vector space basis change: is this “index-free” notation correct?

There are already quite a number of questions on basis change in a vector space. Nevertheless, to fully grasp the underlying idea I made up the following notation and I have some doubts on it (note: ...
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0answers
19 views

A question on existence of a linear map on finite dimensional vector spaces

Suppose that $V, W$ are finite dimensional vector spaces and $U$ a subspace of $V$ such that $\dim U\ge \dim V-\dim W$ , then how do we prove that there is a linear map $T:V\to W$ such that $\ker ...
2
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1answer
40 views

Finding set of vectors that spans the solution set

Question: Find a set of vectors $\{u,v\}$ in $\mathbb{R}^4$ that spans the solution set of the equations: $$\begin{align}w - x + y + z = 0 \\ 5w + 2x - y + z = 0\end{align}$$ Reducing these I get: ...
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1answer
20 views

$\nabla \times \underline{v}$ - Results in a vector perpendicular to these two vectors?

Say $v = -y\hat{i} + x\hat{j}$ If we take the cross product of $\underline{v}$ with $\nabla$ we get $\left| \begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ \frac{d}{dx} & \frac{d}{dy} ...
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4answers
92 views

$(1, 1) \cdot (6, 0) = 6?$ Intuition?

$a = (1, 1)$ $b= (6, 0)$ $a \cdot b = (1, 1) \cdot (6, 0) = 6$ I have seen the dot product of $a$ and $b$ refered to as "What is the x-coordinate of $a$, assuming $b$ is the $x$-axis?". Well here ...
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0answers
11 views

Definition of a Periodic function in vector space

What is the definition of a periodic function in vector space?
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36 views

normal vectors in spaces where $n > 3$

I am reading Lovelock and Rund's book on Tensors and they make a statement that I wanted to validate about normal vectors in high-dimensional spaces. It should be remarked that the above ...
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0answers
29 views

Proving basis of $\mathcal{L}(V,W)$

Suppose two vector spaces $V$ and $W$ over some field $F$ is given. Now let $\mathcal{L}(V,W)$ be the set of all linear maps from $V$ to $W$. Also let $\dim V=n,\dim W=m$ and $\mathscr{M}(m,n,F)$ be ...
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2answers
52 views

General vector space theory developed without matrix-theory.

Since vector spaces can exist regardless of a matrix I wanted to see if we could do all the proofs for the general vector-spaces without using theory for matrices. Then it was only two proofs of the ...
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1answer
47 views

Does $(x,f(x),\cdots,f^p(x))$ is linearly dependent over $E$ implies $(id, f, …, f ^ p)$ is linearly dependent over $\mathcal{L}(E)$?

Here is the original (classic I think) problem I had encored: if $(x,f(x))$ is a linearly dependent family of $E$ (a vector space) for all $x\in E$, then the family $(id,f)$ is linearly dependentt ...
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0answers
30 views

Prove that if $m<n$ then $S$ does not generate $V$

Let $V$ be a vector space over a field $K$ such that $dim V=n$ and let $S\subseteq V$ such that $|S|=m$. Prove that if $m<n$ then $S$ does not generate $V$ Let $S=${$s_1,...,s_m$}. Suppose that ...
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1answer
23 views

definition of linearly dependent set

I know that this is a silly question to ask but I would really appreciate if you can answer me. Let $V$ be a vector space and $S=\{v_1,\ldots,v_n\}$ a finite subset of $V$. $S$ is linearly dependent ...
2
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1answer
20 views

Operation on vector.

i was working in 3d geometry and i looking for know, if a point belongs to a triangle. I found this "Determine if projection of 3D point onto plane is within a triangle" but i doesn't know how resolve ...
2
votes
2answers
36 views

How to figure the size of the following vector set?

Let $V$ be the set of all vectors in $\mathbb R^n$ with entries $±1$. What is the size of this vector set? I know the answer is $2^n$ but I cannot prove why. I feel like this has something to do ...