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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Farey Sequence Vector Orthogonality Relation Question

Take the Farey sequence $\mathcal{F}_n$ for $n=39$ with values $a_m\in \mathcal{F}_n$ and put them into a vector $$ \vec v_k=\biggr(\exp(2\pi i k a_m)\biggr)_m $$ Since Merten's function for $n=39$ ...
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$\operatorname{rank}(F) = \operatorname{dim}_{k}(\frac{F}{mF})$

Let $R$ be a commutative ring with unit; $m$ is a maximal ideal; $F$ a free $R$-module. We know that $\frac{F}{mF}$ is a vector space over $\frac{R}{m} = k$ . I have to prove that ...
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Does the dimension of R have anything to do with proportion of inscribed triangles to their paralellepipeds?

I had a hard time deciding on how to title this question. I hope it gives people some idea so they can give me their two cents. We all know that the formula for a triangle is $$ (1/2)b*h$$ In ...
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Question about Dimension

Suppose that $A$ is the space of all symmetric $n \times n$ matrices with complex entries. I want to find the dimension of $A$. I know that the if the entries were real, the dimension is $ ...
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23 views

Why is tree traversal the fastest ray-box method?

I'm learning ray tracing (the problem of intersecting a ray, aka a vector, against a 3D box defined by a max and a min point) and I'm wondering: why is a tree traversal (e.g. bounding volume ...
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Vector spaces and intersections

I was thinking of the following problem lately: Suppose $V_1,V_2,V_3,V_4$ are vector subspaces of $\Bbb{R}^4$ of dimension $2$ such that $V_i\cap V_j=\{0\}$ for $i \neq j$. Is it true that we can ...
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30 views

Strict subsets in vector space

Find proper subsets $S_1,S_2,S_3 \neq \{(0,0)\}$ in vector space $\mathbb{R}^2$ so that: $$S_1 + S_2 \subsetneq S_1$$ $$S_2 \subsetneq S_2+S_2$$ $$S_3+S_3=S_3$$ I wrote out the addition definitions: ...
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1answer
81 views

How many unit vectors can tile an n-sphere with a given angle?

Given a unit radius $n$-sphere, and a constant $c = cos(\theta)$, $0 \le \theta \le \pi$, what is the size of the largest possible set of unit vectors $U = \{u_1, u_2, ..., u_n\}$ such that $u_i \cdot ...
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58 views

How to find the axis of rotation needed to rotate a $ 3d$ vector to another $3d$ vector?

I have two vectors $(a,b,c)$ and $(d,e,f)$. How can I find the axis of rotation needed to rotate the first vector to be parallel to the other vector? Thanks
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237 views

How to rotate a 3d vector to be parallel to another 3d vector using quaternions?

I have a vector (a,b,c) and another vector (d,e,f). I'm trying to rotate (a,b,c) so its parallel to (d,e,f) using quaternions. I need help understanding how I would do this. I have so far that a ...
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2answers
93 views

Row reduced matrix $\Leftrightarrow$ vectors (rows) are linearly independent.

Let $A$, a row-reduced matrix (after applying Gaussian elimination). Show that all rows which are different from $V_0$ (zero vector), are linearly independent. We learned this as sort of an ...
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48 views

Finding an orthogonal basis outside the intersection of two subspaces.

Let $A_1 \in \mathbb{C}^{10 \times 200}$, $A_2 \in \mathbb{C}^{10\times 200}$ and let $G_1 \in \mathbb{C}^{200\times 190}$ represent an orthogonal basis for $N(A_1)$, $G_2 \in \mathbb{C}^{200\times ...
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99 views

Is there a Taylor series for vector cross product?

I have this equation, where $u,v,w,a,b,Ɵ$ are constants. The RHS comes from the Geometric definition of the LHS $(u,v,w)(a,b,c)=||(u,v,w)||||(a,b,c)||\cos(\theta)$ Expanding the 2-norms ...
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1answer
73 views

Using Cylindrical Coordinates to Compute Curl

I am given a vector field $\vec{A} = A_\rho \space \hat{e_\rho} + A_\phi \space \hat{e_\phi} + A_z \space \hat{e_z}$, and I am supposed to use the unit vectors (provided below) in cylindrical ...
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1answer
56 views

Dual Basis problem

I've been dealing with this but I haven't been able to understand the underlying principles of dual basis, so i don't know how to do it well. It starts like this: Have $(e_1, e_2, e_3)$ basis of the ...
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2answers
150 views

Determine whether the set spans $R^2$

So I'm told to determine whether a set spans $R^2$ and if it doesn't then give a geometric description of the subspace that it does span. $S=\{\left(-1,4\right),\left(4,-1\right),\left(1,1\right)\}$ ...
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3answers
182 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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1answer
32 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 ...
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1answer
63 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} = ...
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1answer
50 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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211 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 ...
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1answer
37 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... ?
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66 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 $C^2$ ...
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60 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 ...
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40 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 ...
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1answer
55 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 ...
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35 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: $(u_i)_{i\in \{1,..,n\}}$ ...
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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 ...
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136 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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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 ...
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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 ...
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82 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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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 ...
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1answer
105 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 ...
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1answer
129 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 ...
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363 views

Finite dimensional subspace of $C([0,1])$

Let linear $S$ be a subspace of $C([0,1])$, i.e., the continuous real-valued functions on $[0,1]$. Assume that there exists $c>0$, such that $\|\,f\|_\infty\leq c \|\,f\|_2$, for all $f\in ...
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1answer
41 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?
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48 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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141 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 ...
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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 ...
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131 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 ...
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5answers
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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: $||x|| = \sqrt{\langle x, x \rangle}$, ...
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2answers
308 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 & ...
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1answer
79 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 ...
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165 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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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 ...
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1answer
51 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 ...
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2answers
99 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 $F$ ?
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200 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 ...
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0answers
55 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 ...