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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12 views

How to understand the meaning of 'Oblivious' in Oblivious Subspace Embedding?

For the definitions of Oblivious Subspace Embedding and Subspace Embedding, please refer to the 1st page of paper http://arxiv.org/pdf/1308.3280v1.pdf.
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15 views

Distance between all rows in 2 matrices expressed as a matrix equation

I have two matrices: $X, Y$ with $X$ being of dimension $n_1$ x $p$, $Y$ of dimension $n_2$ x $p$. All real numbers. The goal is to form the matrix $D$ of dimension $n_1$ x $n_2$ where each element ...
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1answer
65 views

Under which branch of mathematics do vectors fall into? [on hold]

Well... this is pretty basic. Under which branch of mathematics do vectors fall into? A quick search on the web revealed many kinds of vectors so what I have in mind are this kind of vectors: that ...
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1answer
33 views

Verifying a Vector Space Via Given Axioms

Let $X$ be the collection of all sequences $\{\alpha_n\}_{n=1}^{\infty}$ of scalars from $\mathbb{K}$ such that $\alpha_n=0$ for all but a finite number of values of $n$. Define addition and scalar ...
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1answer
18 views

Vector Spaces and Simple Modules

Let $G$ be a finite group and let $R = \textbf{R}[G]$ be the group ring of $G$ with coefficients in the field $\textbf{R}$ of real numbers. Let $V$ be an $R$-module which is finite-dimensional as an ...
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2answers
22 views

Linear transformation with special properties

how should I do that please (I had this in my test yesterday)? Linear transformation $f:\mathbf{R}^{10} \to \mathbf{R}^7$ has an attribute that every vector $\mathbf{v}$ for which is true that ...
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2answers
74 views

Is $\mathbb{R}^2$ the same as my dear $\mathbb{C}^2$?

The question is$$\text{Is }\mathbb{R}^2\text{ a subspace of }\mathbb{C}^2?$$My first thing to think about it now is $$\text{Is }\mathbb{R}^2\text{ a subset of }\mathbb{C}^2?$$ I think no because what ...
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0answers
27 views

Try to prove linear independence

I know that since vectors $v$ and $w$ are linearly independent, $av+bw=0$. Should I continue with the assumption that $v, w, v \times v$ are linearly independent so get $av+bw+c(v \times w)=0$? If ...
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1answer
15 views

Span of a set of vectors

In Artin's Algebra book there is the following Lemma about vector spaces: Let $S$ be an ordered set of vectors of $V$, and let $W$ be a subspace of $V$. if $S\subset W$, then Span $S\subset W$. Now, ...
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1answer
14 views

Proof that the velocity vector is tangential to the path?

In calculus class my teacher asserted that the velocity vector is tangential to the path a point takes. I have tried to prove this but have gotten stuck. I computed $\dfrac{v_y}{v_x}$ to be ...
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1answer
24 views

Is every finite dimensional linear space a banach space [on hold]

Is every finite dimensional linear space a Banach Space? Is every finite dimensional linear space a Hilbert Space?
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20 views

How do I find the common invariant subspaces of a span of matrices?

Let $G_1, \ldots, G_n$ be a set of $m\times m$ linearly-independent complex matrices. Let $\mathcal{G} = \operatorname{span}\left\{ G_1, \ldots , G_n\right\}$ be the vector space that spans the set ...
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34 views

prove the subspace is equal to L1∩L2

Let $L1,L2$ be subspaces of $V$ and $dim(L1+L2)=1+L1 \cap L2$ Show that $L1 \subseteq L2$ or $L2 \subseteq L1$ and $L1+L2= L1$ or $L2$ it's has something to do with if $L1,L2 \ne L1 \cap L2$ ...
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1answer
21 views

Cosine Similarity between two sets of vectors?

I have words represented as vectors, and so I can compare two words using the cosine similarity of each word vector. But, now I'd like to extrapolate that and compare two sentences, each being a set ...
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1answer
27 views

Difference between lines dividing planes and planes dividing space

Let a(n) represent the number of regions that the plane R2 is broken into by n lines (no 2 of which are parallel, and no 3 of which intersect in a single point). Let b(n) represent the number of ...
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32 views

Gradient function

Let A (red) and B (green) 2 distinct points anywhere in a 3D space. I am looking for a function which take a point P, and returns the value in blue in the picture. Each blue number in the picture ...
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1answer
10 views

dimension of an intersection of subspaces

Let $V$ be the vector space of all polynomials in one variable with real coefficients having degree at most 20. Define the subspaces \begin{align*} W_1 &=\{p \in V; p(1)=0,p(1/2)=0, ...
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1answer
28 views

Vector spaces and multiplicative inverse?

Do vector spaces have multiplicative inverses? They seem to be monoids under $+,\times$, so monoids $(\Bbb F, +)$ and $(\Bbb F, \times)$ where $\Bbb F=\Bbb R \,or\, \Bbb C$ And it is even a group ...
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1answer
22 views

How to find the normal vector in a TNB problem

I have done this TNB problem multiple times; however, my online homework system keeps telling me my answer is incorrect. I was hoping someone would look at my work and tell me where I'm going wrong? ...
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1answer
34 views

Finding a linear transformation such that $T^{3} = T $

I have to show that there exists a linear transformation such that $T^{3} = T $ i can see that from here that T has eigen values $0.1.-1$ .But how do i find linear transformation .Also for v and q ...
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1answer
18 views

Length of a complex vector

From the definition of inner product in $\mathbb{F}^n$ $$\textbf{a}\cdot\textbf{a}=\sum\limits_{k=1}^na_{k}\overline{a_{k}}$$ Say $a_{k}=x_{k}+iy_{k}$, then ...
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1answer
18 views

how can I calculate the 4 corners of a finite plane that rests in a 3d space

I have a finite plane in my application. The plane is described by its centre point C, its normal vector N and a scale vector S. S is not really a vector but rather a "convenient container" of scale ...
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19 views

Geometry of Spans in $\Bbb{R}^2$ and $\Bbb{R}^3$

I'm having difficulty figuring out how to approach the following Geometry of Spans questions. I only seem to understand the "span of a single vector" ones. How would I go about explaining the others? ...
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26 views

A question about the vector space of Fibonacci sequences

Question: Let $V$ be the vector space of real sequences over $\mathbb{R}$. If $W$ is the subspace of all Fibonacci sequences (i.e. a sequence $\{a_n\}\in W$ if $a_n=a_{n-1}+a_{n-2}$, for all $n\geq ...
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1answer
24 views

Misunderstanding in the proof that the sum of subspaces is the smallest containing subspace.

So if $V_1,...,V_n$ are subspaces of $M$ then $V_1+...+V_n$ is the smallest subspace of $M$ containing $V_1,...,V_n$ The proof is that clearly $V_1,...,V_n$ are all contained in $V_1+...+V_n$ Then ...
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1answer
26 views

Two sets of polynomials with distinct roots build the ring of polynomials.

Definitions: $i \in K$ $U_{i}:=\{f\in K[X] |f(i)=0 \}$ $K[X]$ is the ring of polynomials HINTS: K[X] is a vector space Every $U_{i}$ is a vector subspace of $K[X]$ Question: (i) With $s \neq ...
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1answer
29 views

Linear Transformations on Infinite Dimensional Vector Spaces

Let $T$ be a linear transformation $T:V\to V$, where $V$ is an infinite dimensional vector space. How can we construct examples such as $1.$ T is one to one but not onto $2.$ T is onto but not ...
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1answer
72 views

The set of differentiable functions such that $f'(2)=b$ is a linear subspace if and only if $b=0$??

Questions are in bold. The set of differentiable real-valued functions on (0,3) such that $f'(2)=b$ is a subspace of $(0,3)\to \mathbb R$ if and only if $b=0$ ($(0,3)\to \mathbb R$ denotes the set of ...
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2answers
101 views

Matrix notation of vectors?

My linear algebra book says that a vector is a one-column matrix. However, in the context of what we are studying (linear equations) it would make more sense if a vector was of the form of the ...
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2answers
33 views

Dimension of $n \times n$ diagonal matrix with characteristic polynomial $(x-a)^{p}(x-b)^{ q}$?

What is the dimension of an $n \times n$ diagonal matrix with characteristic polynomial $(x-a)^{p}(x-b)^{ q}$? Do I have to make distinct cases with as $p + q < n$ and equal to $n$? And if their ...
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3answers
52 views

I have difficulty understanding functions forming vector space.

I have knowledge of basic linear algebra, so I can understand the finite vector space as linear combinations of vectors of $R^n$. However, when it comes function as vector and functions form a ...
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3answers
20 views

Find all linear operators such that $F^2 = F$ and $F(x,y) = (ax,bx+cy)$

I need to find all linear operators that match $F^2 = F$ and $F(x,y) = (ax,bx+cy)$ *where $F^2$ means $F$ composed with itself. So what I did: $F(x,y) = (ax,bx+cy)\implies F(F(x,y)) = ...
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4answers
147 views

Definition of basis

There are something that I am not quite sure about the definition of basis. Let $V$ be a vector space over $K$, then the definition of basis says the vectors $v_1,...v_n$ form a basis of $V$ if they ...
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2answers
65 views

$V$ is finite dimentional over field $K\iff$ field extension $L/K$ is finite

Let $L/K$ be a field extension and $V$ a non-zero vector space over $L$. Prove that: $V$ is finite dimensional over $K\iff V$ is finite dimensional over $L$ and $[L:K]<\infty$ for the first ...
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1answer
39 views

Find a basis for $\mathbb{R} ^5$ containing the given vectors

Find a basis of $\mathbb{R}^5$ that contains the vectors $(1,-1,1,-1,0)$, $(-1,-1,1,-1,0)$ , $(-1,1,1,-1,0)$. I think I need to find two more vectors so that the five vectors are all linearly ...
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0answers
24 views

PDEs, infinite vector spaces, and functional analysis

Suppose I have a PDE for some quantity $q_{t}$:$= q(\boldsymbol{x},t)$ at some time $t$ on a 2D vector space $\boldsymbol{x} = (x_{1},x_{2})$, that looks, for the sake of the question, something like ...
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1answer
20 views

Finding same-vectors that have same coordinates in two different basis

I have two different vector basis: Default: $\{e_1,e_2,e_3\} = \{(1,0,0);(0,1,0);(0,0,1)\}$ Special basis: $\{e'_1,e'_2,e'_3\} = \{(1,1,1);(1,0,1);(0,2,1)\}$ My question is: How do I find which ...
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2answers
42 views

Let $F$ be a linear operator such that $F^2 - F + I = 0$, show that $F$ is invertible and $F^{-1} = I - F$

I didn't understand this exercise. I tried working with $$F^2 - F + I = 0\implies (F-I)(F) + I =0$$ but I really don't understand how to prove $F$ is invertible neither find the inverse. Any hints? ...
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1answer
20 views

Verifying if $F$ such that $F(1,0) = (2,5)$ and $F(0,1) = (3,4)$ is an automorphism

What I did: $$(x,y) = x(1,0) + y(0,1)\implies\\F(x,y) = xF(1,0) + yF(0,1)\implies\\F(x,y) = x(2,5) + y(3,4) = (2x+3y, 5x+4y)$$ I need to verify if $G = I + F$ is na automorphism. So: $$G = I + F = ...
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1answer
11 views

Tetrahedron in vector space: Finding a vector connecting two points

Edited to add: The tetrahedron is not necessarily a regular one. First off, the point $M$ is the centre of gravity for this tetrahedron. I have a base $\{e_1,e_2,e_3\} = ...
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0answers
8 views

Curl of a 2D Spherical Field [on hold]

I have a (r,θ) 2D spherical polar field and I want to find its curl. How would I go about doing this? Thanks in advance.
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2answers
15 views

Rank of a matrix from a 5 X 7 matrix with a basis of 3 vectors

The question in my book is as follows: If the subspace of all solutions of Ax=0 has a basis consisting of thee vectors and if A is a 5 x 7 matrix, what is the rank of A? Now i thought because ...
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1answer
170 views

Prove generalisation of the Tower Law

I need to prove the following generalisation of the Tower Law: Let $L/K$ be an extension of fields, and $V$ a non-zero vector space over $L$. Then $V$ is finite-dimensional over $K$ if and only if ...
2
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1answer
99 views

Estimating rank and nullity of the composition of linear maps

Let $T\colon U\to V$, $R\colon V\to W$ be linear maps between finite dimensional spaces $U$, $V$, $W$, and let dim$(V)=n$. Prove that $\dim\, \ker(RT)\le \dim\, \ker(R)+\dim\, \ker(T)$, ...
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1answer
28 views

Is this set of functions a vector space?

I'm starting to learn linear algebra am an learning what is and what is not a vector space. I'm trying to figure out if the following set of functions is a vector space: {f : R → R | f(3) = 0} I ...
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1answer
41 views

Are functions infinite dimensional vectors? [closed]

Are functions infinite dimensional vectors? There are a few sources on the internet that makes this claim, but they do not cite any sources which makes me feel like they are just using it as an ...
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1answer
29 views

$cu=0\iff u=0\vee c=0$

$c$ is a constant in R or C $u$ is in a vector space $cu=\mathbf 0\iff u=\mathbf 0\vee c=0$ First I tried to show the two implications $cu=\mathbf0\Leftarrow u=\mathbf0\vee c=0$ and ...
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2answers
57 views

Is the empty set a vector space?

I think the empty set satisfies all of the axioms of a vector space except the one about the existence of an additive identity. Is this right?
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38 views

Can these characterisations of finite dimensionality be proven equivalent without using a basis?

I was wondering about how to define "finite dimensional" without talking about bases. Two possibilities occurred to me: Say $V$ is finite dimensional if the canonical inclusion $V\hookrightarrow ...
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1answer
28 views

Conceptual question about the extended real line and being a vector space.

Last time I was chatting with a professor online on a public IRC, this is a transcript: #Professor (16:20:12): So what was your question? #Me (16:20:22): I feel stupid.... #Professor ...