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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22 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
12 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
13 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
22 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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18 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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0answers
31 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
15 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 ...
1
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1answer
18 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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0answers
28 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
9 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
27 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? ...
0
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1answer
33 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 ...
2
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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
14 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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0answers
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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0answers
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
25 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
100 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
145 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 ...
0
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1answer
36 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
23 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 ...
2
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1answer
19 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
41 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
164 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
votes
1answer
96 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? [on hold]

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 ...
1
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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
56 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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0answers
36 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 ...
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1answer
43 views

Is a subset a vector space if it is just one vector?

I'm beginning to learn linear algebra and wanted to know if a subset of a vector space is a vector space if there is only one vector. For example, $$V = \{(x, y, z) \in \mathbb{R}^3| x^2 + y^2 + z^2 ...
7
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2answers
243 views

Finite-dimensional space naturally isomorphic to its double dual?

The example that a finite vector space is naturally isomorphic to its double dual seems to be the canonical example of natural isomorphisms. Concretely, there are two functors $\mathsf{Id}, {-^*}^* : ...
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0answers
21 views

which book and document? [closed]

I want to study about weak topology and weak star topology. So, what can I read books or documents? With 'Functional analysis', which books are good to study?
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1answer
20 views

How to find the transition matrix from basis $E$ to $E'$

Suppose there is a linear transformation $T$ on $\mathbb R^n$. And $$E=[\epsilon_1,\epsilon_2...\epsilon_n]$$and $$E'=[\epsilon'_1,\epsilon'_2,...\epsilon'_n]$$ are two different basis of $\mathbb ...
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2answers
59 views

Prove linear independence of a set $\{\mathbf{x}-\mathbf{x_1},\ldots,\mathbf{x}-\mathbf{x_n}\}$

Let $V$ be a vector space and suppose that $\{\mathbf{x_1},\ldots,\mathbf{x_n\}}$ is a linearly independent subset of $V$. If $\mathbf{x} = \sum_{i=1}^n c_i\mathbf{x_i}$ where each $c_i \in ...
0
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2answers
45 views

Prove that a Polynomial ring is a vector space

so I shall prove that a Polynomial ring [K] is a vector space. How do I do that? I was thinking of just going down all axioms one by one.. but I don't really know how to prove them for a polynomial ...
0
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1answer
32 views

Understanding relation between vector valued function and function objective in an multi objective optimization problem

I try to understand the relation between "vector-valued function" and "function objective" as used in optimization problem. I understand that objective function in a multi-objective problem can be ...