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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$3D$ surfaces multivariable Calculus

A surface is constructed as follows: First a curve $(0, y, −((y − 1)^2)((y + 1)^2))$ is drawn in the yz–plane. Then a parabola $(u, u^2)$ is drawn in the uv–plane. Finally, in each plane y = b, a copy ...
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How can i get a vector of fixed magnitude in a particular direction?

I am programming something and i am stuck with this problem related to mathematics. I am not so good in maths. I have Vector A and Vector B. I know that to get the direction from A to B, I have to ...
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1answer
27 views

Basis of the row-space of a matrix with non-negative entries.

Consider a matrix $A \in \mathbb{R}^{n \times m}$ such that all entries are non-negative. Denote the rank of $A$ as $k$. I am mostly interested in cases where $k \ll n$, but this probably isn't ...
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2answers
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How would I be able to tell if some vector is in the span of a set of vectors?

Given the following, how would I be able to tell if b and c are in the span of the set of vectors S? Any help is appreciated. enter link description here
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1answer
13 views

dimension of quotient space

Let $f(x)=x^4+3x^3-x^2-4x-3$ and $g(x)=3x^3+10x^2+2x-3$ and $U = \{u(x)f(x)+v(x)g(x) | u(x),v(x) \in \mathbb{F}[x]\}$, find the dimension of quotient space $\mathbb{F}[x]/U$ If $V$ is a finite ...
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Tangential and normal components of acceleration acting on a dropped bomb

The original question is this: A plane flying at an altitude of 34,000 feet at a speed of 510 miles per hour releases a bomb. Find the tangential and normal components of acceleration acting on ...
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25 views

Determining whether $\mathbb R^+$ is a vector space under given actions

While $\mathbb R^+=V$ ($\mathbb R$ is the group of all real positive numbers), we define the following actions: $\forall a, b \in V$, $a \oplus b = a \cdot b$ $\forall a \in V$, $\forall ...
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1answer
19 views

Proving properties of orthogonal subspaces

If $M$ and $N$ are subspaces of $\mathbb{R}^n$, show that $$(M+N)^\bot=M^\bot \cap N^\bot$$ and $$(M \cap N)^\bot = M^\bot +N^\bot $$ I found this previous question, but I cannot seem to make sense ...
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1answer
75 views

Quotient Space (W-Affine Subspaces) “Proof” Verification.

Let $(V, K)$ be a vector space and $W ⊂ V$ a subspace. A subset $S ⊂ V$ is called a $W$-affine subspace of $V$ if the following holds $∀s, s ∈ S, s − s ∈ W$ and $∀s ∈ S, ∀w ∈ W, s + w ∈ S$. ...
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2answers
122 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 ...
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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) = ...
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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.
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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 ...
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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 ...
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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 $\mathbb{R}^3$ ...
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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 ...
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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 ...
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1answer
41 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 ...
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0answers
34 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
34 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 ...
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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 ...
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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 ...
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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 $v\circ u\circ v=v$ ? ...
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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 ...
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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 $ ...
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0answers
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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 $\mathbb{R}$. ...
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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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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 ...
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0answers
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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
53 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 ...
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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 ...
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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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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' ...
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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 ...
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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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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, ...
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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 $\angle(BAC)=90°$. ...
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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 ...
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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
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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 ...
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 ...
0
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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.