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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5
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2answers
121 views

Are there norms on $\Bbb{C}^m$ and $\Bbb{C}^n$ so that the norm $\Vert\cdot\Vert$ is a subordinate norm?

Denote $$\Vert A\Vert=\sum_{1\le j,k\le m}\vert A_{j,k}\vert$$ is cleary a norm over $M_{m,n}(\Bbb{C})$ but not a subordinate norm by taking the identity matrix $I$. So my question is: Can we make ...
0
votes
2answers
86 views

Isomorphism of all Linear Transformation

my work book has a questions that asked us to prove something, however the answer was not provided. The question states that: Let $V$ and $W$ be vector spaces over $F$ where the $dim(V) = n, dim(W) = ...
0
votes
1answer
44 views

Show that V is a subspace by expressing it as the span of a set of vectors

What exactly is this question asking me to do? I think the use of the set notation has thrown me off a bit. Any help is appreciated.
-1
votes
0answers
19 views

Basis span of space necessary to be orthogonal?

Q1 If a vector space V that span of {v1,v2,....,vk},can the basis vector of V are not mutually orthogonal? (From several users comments, the answer is the basis vector can be no mutually ...
0
votes
2answers
36 views

Finding a basis for a subspace in $\;\Bbb R^4\;$

I know this might be a really simple question to ask but I just don't understand how to obtain the answer to this question. I've tried to understand subspaces (and even the difference between a space ...
0
votes
1answer
33 views

Are the electric and magnetic fields functions on R^4?

Are the electric and magnetic fields functions from $\mathbb{R}^4$ to $\mathbb{R}^4$ (where $\mathbb{R}^4$ is then interpreted as space-time) or do we consider them to be functions from $\mathbb{R}^3$ ...
0
votes
3answers
32 views

show that dim(L,W) = mn

There are two finitely dimension vector spaces $V$ and $W$. Dimensions are $n$ and $m$ respectively. $$L(V,W)=\{T:V\rightarrow W \;|\; T \;\text{is linear}\}$$ $L(V,W)$ is a vector space with ...
0
votes
1answer
18 views

What can I say about the dimension of all real functions?

If I have a vector space of all real functions And S is all real functions with no constant term. then S is the subspace of V. Then, What can I say about the dimension of S? V has infinite ...
1
vote
1answer
40 views

Why is the infinite dimensional vector space with only finitely many nonvanishing components incomplete?

Define a complex vector space $V$ such that any element $\{a_i\}=(a_1,a_2,\dots)\in V$ has only finitely many components $a_i\ne 0$. The inner product is defined as $$(\{a_i\},\{b_j\})=\sum_i^\infty ...
0
votes
0answers
31 views

Rank Nullity Theorem for Infinite dimensional vector spaces

Rank nullity theorem can be extended for infinite dimensional vector space.Can someone help me to complete my proof.I think this idea will work. Rank Nullity Theorem states that if $T$ is a linear ...
0
votes
1answer
18 views

Proving the 0 Vector and the Set of Eigenvectors of a Linear Map are a Subspace

Given the map T∈ L(V) where L(V) is the set of all linear maps from V to V. I'm wondering whether it can be proven that the set of {the 0 vector and all the eigenvectors of T} can be shown to be a ...
0
votes
1answer
29 views

Easy question about vector spaces

Suppose $F$ is a (added later: finite-dimensional) vector space over $K$ and $K'$ is a subfield of $K$. If $\dim_K F = \dim_{K'} F$, then how does one prove that $K=K'$? Somehow I can't quite show ...
0
votes
0answers
39 views

Existence of a basis in constructive vector spaces

As I was trying to review forgotten knowledge on Vector Spaces in wikipedia, I read that the existence of a basis follows from Zorn lemma, hence equivalently from the axiom of choice. Actually, the ...
0
votes
0answers
58 views

Find 2 unit vectors that make an angle of $60^\text{o}$ with $\vec v=\langle 3,4 \rangle$

Find 2 unit vectors that make an angle of $60^\text{o}$ with $\vec v=\langle 3,4 \rangle$. My working: $$\cos{60^\text{o}}=\frac{1}{2}= \frac{\langle u_1,u_2\rangle\cdot\langle ...
0
votes
1answer
39 views

3D plane rotation about a line

In three dimensional space we have a plane and a line. These can be oriented in any way. The plane is rotated about the line by n degrees, meaning that originally the position of the plane is fixed to ...
1
vote
1answer
58 views

Is there a difference between $a \cdot a^T$ and $a^2$?

The title says it all... I can't see the difference between $a \cdot a^T$ and $a^2$, when $a$ is a vector. However I encountered a formula stating $$\frac{1}{|y+a|} = \frac{1}{|y|} - \frac{y \cdot a ...
0
votes
2answers
39 views

Find a normal vector onto the line

How can I find normal vector on the given line. For example if I have a line $3x - 5y = 1$, what would be the normal vector of this line? I am not sure whether it's useful or not, but we have one more ...
1
vote
1answer
33 views

Find a basis for a subspace (working included)

I have been working on this question and I am not too sure if it is correct or not. Any help would be appreciated. Question (in picture format): http://i.imgur.com/E4MhH99.png My working: The first ...
-1
votes
1answer
25 views

Proof that R2 belongs to (a+b, b) [duplicate]

I am aware that the vector (a+b, b) belongs to R2 for a,b being real numbers. Also, I am aware that R2 belongs to (a+b,b), but I am not sure how to prove it. R2 is defined as (x1, x2), x1,x2 are real ...
1
vote
2answers
58 views

Proof that $(a+b, a)$ belongs to $\mathbb{R}^2$

I am aware that the vector $(a+b, a)$ such that $a$, $b$ are real numbers belongs to $\mathbb{R}^2$, which is defined by any vectors $(x_1, x_2)$ such that $x_1, x_2$ are real numbers. Is there a way ...
0
votes
2answers
35 views

$v\in\mathcal{L}(F,E)$ such that $u\circ v\circ u=u$

Let $E,F$ two $\mathbb{K}$ vector spaces, $u\in\mathcal{L}(E,F)$. a) Show that there exists $v\in\mathcal{L}(F,E)$ such that $u\circ v\circ u=u$ b) Can we additionally have $v\circ u\circ v=v$ ? ...
0
votes
1answer
32 views

Understanding the operator of differentiation on the vector space of polynomials

I have been reading through Linear Algebra Done Right by Sheldon Axler. The book defines an operator as a linear map from a vector space to itself. It then considers at another part of the book the ...
1
vote
2answers
46 views

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 $ ...
-1
votes
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 $\mathbb{R}$. ...
0
votes
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 ...
2
votes
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 ...
-3
votes
0answers
23 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 ...
0
votes
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 ...
0
votes
2answers
52 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 ...
2
votes
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 ...
0
votes
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 $ ...
0
votes
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
votes
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 ...
1
vote
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 ...
0
votes
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, ...
0
votes
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 $\angle(BAC)=90°$. ...
1
vote
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 ...
1
vote
1answer
48 views

Tait-Bryan to Rotation matrix to translating from global to local space

Re-writing my entire question to be more math-oriented and hopefully make more sense. I have two objects, each at a position defined by P1 and P2 (XYZ). Each has a heading based on yaw/pitch/roll, ...
1
vote
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 ...
0
votes
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 ...
0
votes
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
vote
1answer
22 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
votes
1answer
17 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 ...
0
votes
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 $\binom{x}{k}$ ...
1
vote
0answers
20 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? $\displaystyle y=\frac{cos(s_n)-cos(s_{n+1})}{sin(s_{n+1}-sin(s_n)}(x-sin(s_n))-cos(s_n)$
1
vote
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
vote
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
votes
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 ...
0
votes
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 ...
1
vote
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 ...