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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61 views

Proof about finite dimensional vector spaces over fields

Prove that every finite dimensional vector space $V $of dimension $n$ over a field $F$ is isomorphic to the vector space $F^n$. Okay, lot's of stuff here. I think most of the reason I can not do this ...
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0answers
13 views

Sequence of sets and vectors

Let $A(0,1),B(0,0),C(1,0)$ and $D(1,1)$ be four points in the plane $xOy$. Define $M_3=\{A,B,C\}$ and $M_{n+1}=M_n \cup \left\{Z\epsilon xOy\mid \exists V,W\epsilon M_n\text{ for which ...
3
votes
1answer
471 views

How to plot N points on the surface of a D-dimensional sphere roughly equidistant apart?

Let's say I have a D-dimensional sphere with a radius R. I want to plot N number of points evenly distributed (equidistant apart from each other) on the surface of the sphere. It doesn't matter where ...
5
votes
3answers
189 views

Why orthogonal basis?

Lets take the $\mathbb{R}^3$ space as example. Any point in the $\mathbb{R}^3$ space can be represented by 3 linearly independent vectors that need not be orthogonal to each other. What is that ...
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2answers
38 views

How do you prove that every vector enclosed in a space has a unique linear combination made from the basis of the space?

Let's say there's a vector $\mathbf{v}$ in space $\mathbf{V}$, and the basis for $\mathbf{V}$ is given by $\mathbf{S}=\{v_1, v_2, ... v_n\}$. I start out with an equation $A \mathbf{x} = ...
0
votes
1answer
34 views

Overlapping null spaces

Let $A\in \mathbb{C}^{N\times M}$, $B\in \mathbb{C}^{N\times M}$, $M>N$. $\dim(N(A)) = \dim(N(B)) = M-N$. Obviously the null spaces intersect: $$\dim(N(A)\cap N(B)) = ...
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2answers
53 views

Why I need to study Matrix and Vectors in maths

I am presently learning C and C++ programming. I want to make my profession as a C and C++ programmer. Well. In Data structure concepts, I can see lot of matrix material. In school time, I used to ...
1
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1answer
27 views

when subtracting two vectors, does it matter which one you subtract from what?

If I have two vectors AB and CD, is saying CD - AD the same as AD - CD? If not, what will be the difference in the two resulting vectors? Direction only, or... ?
1
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0answers
45 views

Real version of the Jensen's formula.

Prove the Jensen's formula $$\int_{T}f(z+re^{2\pi i\theta})d\theta-f(z)=\iint_{D(z,r)}\log{\frac{r}{|w-z|}}\Delta f(w)dm(w)$$ where $w$ is in $D(z,r)$ and $f$ is a two-dimensional $C^2$ ...
1
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2answers
217 views

Prove Convex Hull of Minkowski sum

I want to prove that the following holds, where the $+$ means Minkowski sum: $$ conv(A+B)=conv(A)+conv(B) $$ Let the convex hull of $A+B$ be $$ conv(A+B)=\sum_{j,k}\lambda_j\mu_k(a_j+b_k) $$ I ...
3
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0answers
45 views

What extra assumption makes this transformation affine?

Let a vector space $V$ be given. Let $f:V\to V$ have the property that for all $x,y,a\in V$, $$ f(x+a)-f(y+a) = f(x) - f(y) \tag{$\star$} $$ Q1. I'd like to know how weak one can make additional ...
1
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0answers
29 views

Want to determin a dual space

I would like to determine the dualspace of some normed vectorspace. Namley, $$c_0:=\{x=(x_n)_{n\mathbb N}\subset\mathbb R:\lim_{n\rightarrow\infty}x_n=0\; \text{ and } ||x||=\sup_n|x_n|\}$$ I ...
2
votes
1answer
47 views

Is a normed $\mathbb R $ vectorspace complete in general?

I would like to finde out if a normed $\mathbb R$-Vectorspace is complete in general. Or even in a more general case if a normed $K$-Vectorpace, where K is a close field is complete? I somehow think ...
1
vote
2answers
74 views

Determine whether S is a subspace of P3. Vector space of all real polynomials.

ATTEMPT: Have given a small attempt just really confused on how to approach. So I got the general equation of $p(x)= a + bx +cx^2 +dx^3$. So we find the derivative? and find the values of ...
0
votes
1answer
23 views

Set of vectors linearly independent

Supposing $u_1,u_2,...,u_n$ a set of n vectors of $\mathbb{R}^d$. We define the vectors $v_k=u_1+u_2+...+u_k$, as $k$ is an integer from $1$ to $n$. How can I prove that: $(u_i)_{i\in \{1,..,n\}}$ ...
0
votes
2answers
315 views

Volume of a Pyramid Linear Algebra

Find the volume of a pyramid with triangular base bounded by vectors (1,-1, 2) and (1, 1, 1) and vertex located at (3, 2, 5). I am not sure how I would solve this. I know the volume of a pyramid is: ...
5
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0answers
87 views

Intuition about structures in Galois Theory.

Background Often in mathematics I find we can ask "why" something is true. Of course, this is not a well defined question. However, it usually prompts answers that includes words like ...
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1answer
38 views

matrices forms a basis for vector space 2x2

$\begin{bmatrix}0&1\\2&3\end{bmatrix}$ $\begin{bmatrix}3&4\\5&6\end{bmatrix}$ $\begin{bmatrix}7&8\\9&10\end{bmatrix}$ $\begin{bmatrix}11&12\\13&14\end{bmatrix}$ Show ...
3
votes
1answer
55 views

Show that a normed Vector space is complete, need smart help.

I want to show that a normed vector space is complete. I know that if you can show that every Cauchy sequence converges, then it is complete. But in a normed vector space, completeness is equivavlent ...
2
votes
2answers
28 views

Is there a name for the set of bit combinations of bitstrings?

Let $A \subset \{0,1\}^n$ be a set of $n$-bit bit vectors. Let me call a bit vector $b = (b^{(1)}, b^{(2)}, \dotsc, b^{(n)}) \in \{0,1\}^n$ a "bit combination" of the vectors in $A$ if: $$\forall i ...
4
votes
5answers
2k views

What is the difference between metric spaces and vector spaces?

Does a metric space have an origin? That is, does it have $(0,0)$. Does a vector space have an origin? It seems whatever you can do in a metric space can also be done in a vector space. Is this ...
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2answers
20 views

A set of n generators of a subspace of dimension n.

A set of $n$ linearly independent vectors in $n$-dimensional subset $V$ IS a basis of this subspace. But what about a set of $n$ generators in this subspace? Is it a basis of $V$ for sure?
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1answer
21 views

A generating subset of a vector space contains a basis

Let $V$ be a vector space having dimension $n$, and let $S$ be a subset of $V$ that generates $V$. Prove that there is a subset of $S$ that is a basis for $V$. (Be careful not to assume that $S$ is ...
12
votes
5answers
1k views

Why is one proof for Cauchy-Schwarz inequality easy, but directly it is hard?

Let's say you are in $\mathbb{R}^n$ and you define the norm as $||x||=\sqrt{x_1^2+x_2^2...+x_n^2}$. This we recognize as the usual norm from the inner product: $||x|| = \sqrt{\langle x, x \rangle}$, ...
2
votes
1answer
50 views

Where is piecewise dirichlet function with $|x|^2$ continuous or differentiable?

If $|x|^2$ is continuous and differentiable on all of $\mathbb{R}^n$ (already shown differentiability by showing all $n$ of its partial derivatives are continuous), then... Question: For the function ...
0
votes
1answer
30 views

How to show a subset doesn't span a space?

Given that $\{v_1,…,v_m\}$ is linearly independent, how do you show that $\{v_2,…,v_m\}$ does not span that same vector space?
1
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2answers
43 views

Is $S$ a subspace of $V$?

Let $V$ be the set of real-valued continuous functions on the interval $[-3, 3]$. $S$ is set of real-valued functions with condition $f(-1) = f(1)$. Is $S$ a subspace of $V$? Prove, and if not, why?
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0answers
17 views

How to find a resultant vector for given multiple vectors lie on different position.

I have modelled rectangular features and then I compared my model with a reference model. (Assume everything is in 3D space) . In the below figure, Reference model is shown in light pink ...
0
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0answers
9 views

Cross Product of Covectors

Is the vector/cross product defined for covectors (in the dual space) or is it, strictly speaking, only defined for vectors themselves? I would imagine that it works fine for covectors but I wanted to ...
1
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0answers
28 views

Norm inequality with wedge product

Anyone could help me to prove this following inequality? $\displaystyle\frac{||(u+v)\wedge w||}{||u+v||}\le \frac{||u\wedge w||}{||u||} +\frac{||v\wedge w||}{||v||} $ where $u\wedge v$ is the wedge ...
0
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0answers
39 views

Homework excercise, completeness in Vector-spaces, is it correct?, long, but can it be simplified?

I have a very difficult excercise. I see now that it became too much text for someone to might go through it, if you can please help me, but don't want to read all, can you please then only answer my ...
1
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2answers
50 views

Hilbert vs Inner Product Space

What is the difference between a Hilbert space and an Inner Product space? They both seem to be defined as simply a vector space equipped with an inner product. Also can a metric always be defined ...
0
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0answers
57 views

Zero dimensional ideals and their primary decomposition

Let $S=k[x_1,\dots,x_n]$ be a polynomial ring over a field $k$, and $I$ a zero dimensional ideal with a primary decomposition $I=\cap Q_i$. Why is $\sum \dim_k S/Q_i = \dim_k S/I$?
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0answers
24 views

Shooting a grenade - angle throwing

In my 3D simulation/game I need to shoot a grenade from a grenade launcher. The movement of the grenade is already setup by someone else. all I need is to give him the pitch angle of the grenade ...
2
votes
2answers
38 views

Intersection of two subspaces in 4D

I would like to know if there is some way to imagine the case when a 3D subspace intersects with a 2D plane in a 4D space. For example, let's have a 3D space in 4D $$A = \left(\begin{array}{c}1 & ...
1
vote
1answer
47 views

What is the relation between basis vectors of a vector space to those of its subspace?

From this question: Suppose $V$ is a vector space with dimension $6$. Let A and B be subspaces of V with dimensions 4 and 5 respectively. What are the possible values for the dimension of A ...
0
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0answers
26 views

Books which explain vector analysis/algebra in detail.

I'm trying to learn vectors but I can't find a decent book which explains vectors in depth. I need a book which explains vectors from the beginning, using a beginner's approach(assuming the reader ...
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votes
1answer
32 views

How to use geometry to express unit vectors of spherical coordinate system in terms of Cartesian unit vectors

It's quite easy to express unit vector $\hat{r}$ in sum linear combinations of Cartesian unit vectors $\hat{x}$, $\hat{y}$ and $\hat{z}$. But I am not sure how I can use geomtery to find a Cartesian ...
3
votes
1answer
38 views

Vector space basis

If I have no fundamental misunderstanding of vector spaces, my question is as follows. If an orthogonal basis of a vector space consists of $N$ vectors, is this right that every vector from this ...
2
votes
2answers
40 views

Cartesian & Tensor Product

What is the difference between a cartesian product and tensor product of two vector spaces $V_1$ and $V_2$ defined over same field $F$ ?
6
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2answers
116 views

Question about Normed vector space.

Here is the definition of a normed vector space my book uses: And here is a remark I do not understand: I do not understand that a sequence can converge to a vector in one norm, and not the ...
3
votes
0answers
33 views

Find closest vector to a given vector from a particular set of vector

Let $x=\left(x_t\right)_{t=1}^n$ be a vector such that $$ x_t = \prod_{i=1}^t u_i, \tag{1} $$ where each parameters $u_i$ can take any of two value $$ u_i \in \left\{a,b \right\} = \left\{ 1.3, 0.8 ...
1
vote
1answer
52 views

If $U$ is a subspace of $V$, there exists $W$ such that $T:V\to W$ has $ker(T)=U$.

I am having trouble working out a proof for this question, is it something to do with $U$ and $W$ being complementary subspaces? I cannot find a way to prove that there will always exist a $W$ for all ...
1
vote
2answers
45 views

Existence of a linear transformation in an infinite dimension vector space.

If $V$ and $W$ are vector spaces, $\beta=\{v_1, \ldots , v_n\}$ is a finite a basis for $V$ and $\{w_1, \ldots , w_n\}\subset W$, we know there is an unique linear transformation $T:V\rightarrow W$ ...
6
votes
3answers
229 views

Are $\mathbb{C} \otimes _\mathbb{R} \mathbb{C}$ and $\mathbb{C} \otimes _\mathbb{C} \mathbb{C}$ isomorphic as $\mathbb{R}$-vector spaces?

Are $\mathbb{C} \otimes _\mathbb{R} \mathbb{C}$ and $\mathbb{C} \otimes _\mathbb{C} \mathbb{C}$ isomorphic as $\mathbb{R}$-vector spaces? I am having a very hard time at digesting tensor products ...
1
vote
1answer
28 views

how to do these vectors? need more details

Let $v_1$, $v_2$, $v_3$ be mutually orthogonal non-zero vectors in 3-space. So, any vector $v$ can be expressed as $v=c_1v_1+c_2v_2+c_3v_3$. (a) Show that the scalars $c_1$, $c_2$, $c_3$ are given by ...
0
votes
1answer
70 views

paper about linear independence in altered Vandermonde and Cauchy Matrices

Both Vandermonde and Cauchy matrices with $n$ rows and $k$ ($n \geq k$) columns have the property that any $k$ rows are linearly independent (assuming the coefficient are independent). It seems to me ...
3
votes
2answers
147 views

if $A$ is Abelian group , $B$ is subgroup of $A$ , Is $B \times A/B \cong A$? [duplicate]

If $A$ is abelian group and $B$ is a subgroup of $A$, $B$ is normal subgroup of $A$. Is it true that $B \times A/B \cong A$? I ask because I was watching an online lecture from a course in abstract ...
3
votes
1answer
48 views

Finding the dimension of a vector subspace

Consider $\mathbb{F}_{2}^{n} = \{(k_{1}, k_{2}, ... , k_{n}) : k_{i} \in \{0,1\}$ mod $2\}$. Let $M$ be the subset of $\mathbb{F}_{2}^{n}$ given by $k_{1} + k_{2} + \cdots + k_{n} = 0$. Prove that ...
0
votes
2answers
47 views

Do four dimensional vectors have a cross product property? [duplicate]

we know how to make cross product of three dimensional vectors. $$ \vec A \times \vec B = \vec C$$ where : $ \vec A = (A_i; A_j; A_k)$ $ \vec B = (B_i; B_j; B_k)$ $ \vec C = (C_i; C_j; C_k)$ $C_i = ...