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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Finding linear independence in $v_1,\ldots,v_m$

First, I'll try not to ramble, although it tends to happen when I type. I have the following linear algebra problem for my homework. Prove or give a counterexample: If $v_1, v_2, \ldots , v_m$ are ...
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mean value theorem for scalar field

I just want to make sure that Mean value theorem for scalar field works same as one- dimensional mean value theorem. Usually, my book explains Mean value theorem for scalar field on interval [0.1]. ...
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Mean value theorem and scalar field proof

Assume that f′(x;y)=0 for every x in some n-ball B(a) and for every vector y. Use the mean value theorem to prove that f is constant on B(a). And if f′(x;y)=0 for a fixed vector y and for every x in ...
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Can anyone help me prove derivative of scalar field using mean value theorem?

Assume that f′(x;y)=0 for every x in some n-ball B(a) and for every vector y. Use the mean value theorem to prove that f is constant on B(a). And if f′(x;y)=0 for a fixed vector y and for every x ...
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There is no scalar field such that $f '(a)>0$ for fixed $a$ and for every nonnegative vector $y$ [on hold]

I am trying to prove this. But can't think of how I should start. Anyone has some ideas? and why is there a scalar field $f'(a)>0$ for every $a$ and for fixed vector $y$ ? can anyone give me an ...
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1answer
25 views

Trying to understand proof that 3 non-collinear points determine a unique plane

$Q,R,P$ are 3 non-collinear points. Plane $M = P + s(Q-P) + t(R-P)$. Let $C = Q-P$ and $D= R-P$. Let us grant that C and D are linearly independent. Let $M' = P + sA + tB$. Assume $M'$ has $P,Q,R$. ...
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1answer
21 views

Help to understand the basis for a dual space

I've been introduced to the concept of dual space in linear algebra. I can understand perfectly that the dual space of the space $V$ is a space $V^*$ made of all possible linear maps from $V$ to ...
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1answer
33 views

Can there be ever a counterexample to this?

Does addition on subspaces have an additive identity? I said yes because subspaces are vector spaces, so they must have an additive identity. Which subspaces have additive inverses? I said all of ...
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4 views

Finding rotation axis and angle to align two 3D vector bases

I have asked this question before and, while the accepted answer solved my problem back then, I am still interested in finding the rotation axis and angle. Let me rephrase the problem here: I would ...
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1answer
15 views

Field extension of a vector space

If $V$ is a vector space over the field $k$, and $K$ is a field extension of $k$, then $(V)_K$ over $K$ is a vector space. How this new vector space is constructed? and how are the linear ...
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27 views

Prove that $U_1\cup U_2$ is a subspace of $V$ $\iff$ $U_1\subseteq U_2$ or $U_2\subseteq U_1$ $\triangle$

Let $V$ be a vector space over some field. Let $U_1$ be a subspace of $V$. Let $U_2$ be a subspace of $V$. Prove that $U_1\cup U_2$ is a subspace of $V$ is equivalent to $U_1\subseteq U_2$ or ...
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8 views

Dimension of sum of permutations of tensor products of vector spaces

Sorry for the mouthful of a title! Suppose I have two finite vector spaces $W,V$ with bases $\{w_1\dots w_p\}$ and $\{v_1\dots v_q\}$. Consider some subspace $S$ of $W\otimes V$ of dimension $m$ ...
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15 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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2answers
48 views

Lattice of integers $\mathbf{Z}$ in $\mathbb{R^2}$

Lattice of integers $\mathbf{Z}$ in $\mathbb{R^2}$ The questions: Give an example of a nonempty subset of $\mathbb{R^2}$ (noted $M$) which is closed under addition and for all $m\in M$ we have $-m\in ...
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1answer
67 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
38 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
23 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
77 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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28 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
15 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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22 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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35 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
22 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
45 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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33 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
23 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
22 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
19 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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28 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
27 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
75 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
103 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
53 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
148 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
66 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
21 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? ...