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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What is wrong with this proof that if $V = U_1 \oplus W$ and if $V = U_2 \oplus W$, then $U_1 = U_2$?

Claim: Let $U_1, U_2$ and $W$ be subspaces of a vector space $V$. Suppose $V = U_1 \oplus W$ and $V = U_2 \oplus W$. Then $U_1 = U_2$. "Proof" Let $v \in V$. Then $\exists \space u_1 \in U_1 $ ...
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2answers
29 views

Product over a vector space

When looking at the definition of a vector space, I see that it's basically a set with two operations and a set of 8 axioms. However, none of those axioms talk about the product of two vectors. Is ...
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0answers
21 views

Function in L1 space but not also in L2 space [duplicate]

For example, the function $f(x) = \frac{\sin{x}}{x}$ is in L$_2$ space, i.e. it's square-integrable over $\mathbb{R}$, but it isn't in L$_1$ space, i.e. it isn't integrable over $\mathbb{R}$. ...
2
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1answer
20 views

Computing intersection of vector spaces spanned by two lists

Assume that I'm given two lists of vectors $l_1$ and $l_2$, where all the vectors have equal dimension. I want to compute a basis for the intersection of their spans. What is the easiest setup for ...
2
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1answer
38 views

Problems on vector spaces

Let $E$ a $\mathbb{K}$-vector space of finite dimension $n$, $\mathcal{V}$ a subspace of $\mathcal{L}(E)$ such that $$\forall u\in\mathcal{V}\setminus \{0\},u\in\mathcal{GL}(E)$$ a) Show that ...
1
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1answer
22 views

For a linear function, the fiber of the output is the translate of the kernel by the input. (Trivial observation, proof needed.)

As you may already know, I am a newbie to linear algebra. I am supposed to prove that for every linear function between vector spaces, for every input, the fiber of the corresponding output equals the ...
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1answer
23 views

Function from one Null space to Another

Suppose a single vector space over $R$ of degree $n$, and two matrices $A, B$ of arbitrary row size, but col size $n$, s.t. their individual null spaces are linear subspaces of this vector space. Is ...
3
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1answer
253 views

Vector Project onto Subspace

So the question is: Let S be the subspace of $\mathbb{R}^3$ spanned by the vectors $ u_2 = \begin{pmatrix} \frac{2}{3}\\\frac{2}{3}\\\frac{1}{3}\end{pmatrix} u_3 = \begin{pmatrix} ...
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0answers
22 views

dimension of intersection of two subspaces [closed]

$w_1=${$(0,x_2,x_3,x_4,x_5)\hspace{0.1in} | \hspace{0.1in} \forall x_i \in \mathbb{R} \hspace{0.1in} i = 2,3,4,5$ } & $w_2=${$(x_1,0,x_3,x_4,x_5)\hspace{0.1in} \vert \hspace{0.1in} \forall x_i ...
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2answers
36 views

What does 'dimension' strictly mean?

Ask a simple question but confusing me. Case 1. Take an Eucildean space R^3 for example. R^2 is one of its subspce with bases [1,0] and [0,1], and the dimension of this subspace is 2. So for example ...
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2answers
51 views

Solution to homogeneous linear differential equation form a vector space

Show that the solutions of a homogeneous linear differential equation $y"+a(x)y'+b(x)y = 0$ form a vector space. What is its dimension? I understand that the dimension is 2 and that 0 is a solution ...
0
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1answer
34 views

Subsets that are also vector spaces

The vector space $R^3$ and the subset M consists of the vectors $(\xi_1,\xi_2,\xi_3)$ for which i) $\xi_1 = 0 $ ii) $\xi_1 = 0$ or $\xi_2 = 0 $ iii) $\xi_1 + \xi_2 = 0 $ iv) $\xi_1 + \xi_2 = 1 $ ...
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0answers
15 views

Divergence Theorem coming in different forms

Can someone show me how divergence theorem gives the following three identities?: $\int_S d\textbf{S}'\cdot \textbf{P}(\textbf{r}') \frac{\mathbf{r-r}'}{|\mathbf{r-r}'|^3} = \int_V d^3r' ...
0
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1answer
31 views

Linear Algebra Vector Tracing

Let $A(2,-1,1)$, $B$ and $C$ be the vertices of a triangle where $\overrightarrow{AB}$ is parallel to $\vec{v}=(2,0,-1), $$\overrightarrow{BC}$ is parallel to $\vec{w}=(1,-1,1)$ and $\angle(BAC)=90°$. ...
2
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1answer
52 views

Proving something is a matroid

I am taking a matroid theory class, and I am having trouble understanding an example we did in class: Let $F$ be a field, $E$ a ground set, and $V$ a vector space over $F$. Let $\phi : E ...
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2answers
70 views

Is it ever correct to say that $\vec{a}-\vec{a}=0$?

My textbooks define $$\begin{cases}0\cdot \vec{a}=\vec{0}\\(m+n)\vec{a}=m\vec{a}+n\vec{a}\end{cases}$$ Therefore, $\vec{a}-\vec{a}=(1-1)\vec{a}=0\cdot\vec{a}=\vec{0}$. But is it ever acceptable, ...
1
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1answer
43 views

Finding a unit vector orthogonal to vectors $a$ and $b$

If I understand correctly, the cross product of vectors $a$ and $b$ is orthogonal to both $a$ and $b$. So for an assignment I have to find two unit vectors orthogonal to vector $a = \langle 1,0,4 ...
0
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0answers
53 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 $\binom{x}{k}$ ...
5
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1answer
135 views

Product of binomial coefficient as a basis

I am stuck with the following problem. Every polynomial of degree $d$ can be expressed as $$ p(x) = p_d \binom{x}{d}+ p_{d-1}\binom{x}{d-1} + \cdots + p_0 \binom{x}{0} $$ What is the ...
0
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1answer
62 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 ...
1
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1answer
40 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 ...
1
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1answer
32 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 ...
1
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0answers
19 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? $\displaystyle y=\frac{cos(s_n)-cos(s_{n+1})}{sin(s_{n+1}-sin(s_n)}(x-sin(s_n))-cos(s_n)$
5
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3answers
206 views

What “is” a matrix in the context of a vector space?

I'm familiar with the definition of a vector space $V$ over a field $F$ I'm also comfortable with the notion that a matrix "represents" a linear map from one vector space $V$ to another vector space ...
9
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1answer
267 views

A conjecture about vector space

Let $V$ be a $(r+1)$-dimensional vector space, and $p$ be a positive integer and $1\leq p\leq r-1$. Let $$X=\{v_1,\cdots,v_{2r+1-p}\}\subseteq V$$ be a finite set containing $(2r+1-p)$ different ...
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1answer
19 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.
1
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1answer
20 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: ...
0
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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 ...
1
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0answers
35 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.) ...
0
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1answer
31 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
17 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
20 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 ...
4
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1answer
89 views

Is $\mathrm{col}(\lambda I_n-A)\subseteq \mathrm{col}(B) $ for a complex $\lambda$?

Let $A\in\mathbb{R}^{n\times n}$, let $I_n$ denote the identity matrix of order $n$, and let $ \mathrm{col}$ denote column space. I'm interested in understanding for what values of $\lambda \in ...
2
votes
1answer
222 views

How to recover three successive rotations of a vector

I have a vector, which I rotated with respect to $x$, $y$ and $z$ axes, respectively. Now I want to recover this operation, that means I want to bring it to the previous position by rotating it with ...
0
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1answer
22 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 $V/W$
2
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0answers
160 views

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: ...
0
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2answers
62 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 $S,T \in L(V,W)$ ...
0
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0answers
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
40 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
39 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 ...
2
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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 ...
1
vote
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
43 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 ...
5
votes
1answer
74 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 ...
3
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1answer
55 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 ...
1
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1answer
34 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 ...
0
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1answer
22 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 ...
0
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0answers
27 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. ...
2
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0answers
33 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 ...