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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19 views

Existence of linear/affine subspace for a number of vectors

Let $V$ be a vector space over a field $K$. Let $k \le \dim V$ be a natural number. I want to show that for each k vectors $v_1, ..., v_k$ there is a linear subspace $U$ of $V$ which has dimension ...
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3answers
39 views

Why does linearly independent spanning set imply minimal spanning set for a vector space?

Suppose β is a linearly independent spanning set of some vector space V. Why must it be the minimal spanning set? In other words, why can there not be two linearly independent spanning sets of a ...
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2answers
133 views

Effect of doubly stochastic matrix on vector norm

Let $D$ be a $N \times N$ doubly stochastic matrix, $x$ be a $N$ dimensional vector. What is the relation between $\Vert Dx \Vert_2$ and $\Vert x \Vert_2$? In addition if $\Vert x \Vert_2=1$, what ...
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2answers
21 views

P3 being subspace of vector space?

V = P3 (all real polynomials of degree at most 3) and $S = \{p(x)\in P_3 | x·p'(x) = p(x),\} $ is it a subspace of vector space $V$? Solution: I don't even know is it possible for the equation ...
5
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3answers
320 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 ...
21
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4answers
558 views

Proving an integer is non-negative by showing there is a vector space with it as its dimension.

The other day I attended a lecture on methods to show whether or not a number is an integer. We were given examples of showing it is the number of ways to count something, and to show there exist ...
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2answers
50 views

Difference between Euclidean space and vector space?

I often hear them used interchangeably ... they are very complicated to make any use of. Wikipedia words: Euclidean space: One way to think of the Euclidean plane is as a set of points ...
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1answer
60 views

Show that Z cannot be turned into a vector space over any field. [duplicate]

Show that Z cannot be turned into a vector space over any field. So, we have 2 cases here. Case 1:lets suppose the charF=P, n does not equal 0, then (1+1+...+1)n=1n+1n+...+1n=n+n+...+n=pn=wchich ...
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1answer
20 views

fields and subspaces

Let F be a field and let V=F^F, which is a vector space over F. Let w be the set of all functions f element of V satisfying f(1)=f(-1). Is W a subspace of V? a. Has the zero vector b. closed under ...
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2answers
121 views

Relationship between the singular value decomposition (SVD) and the principal component analysis (PCA). A radical result(?)

I was wondering if I could get a mathematical description of the relationship between the singular value decomposition (SVD) and the principal component analysis (PCA). To be more specific I have ...
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2answers
18 views

dimension of the vector space using matrices

Let $C$ be an $n \times n$ real matrix. Let $W$ be the vector space spanned by $\{I, C, C^2, \ldots C^{2n}\}$. The dimension of the vector space $W$ is $ 1.\ 2n \hspace{4cm} 2.\ \text{at ...
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1answer
32 views

problems with finding a basis

Given is : $\mathbb{R^\mathbb{R}_f}:=\{ \alpha:\mathbb{R} \longrightarrow \mathbb{R}| \alpha(x)=0, \}, \alpha(x)\ne0$ only at finitely many points.Show that: $\mathbb{R^\mathbb{R}_f}$ is a subspace ...
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1answer
52 views

difficulties with prooving: K is a vector space over Z/pZ

I am trying to solve the followong exercise: Given is K as a field with finitely many elements. i) show that K is a vector space over $\mathbb{F}_p:=\mathbb{Z/p\mathbb{Z}}$, for some special values ...
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1answer
13 views

V- vector space, show the following equations…

Let V be a K-vector space and S,T $\subseteq$ V be any subset. a. Prove the equation $ <S \cup T>=<S>+<T>$ b. Show based on a counter-example proof that the equation $ <S \cap ...
3
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1answer
23 views

Prove is linearly independent

Prove that that the following subset $S \subseteq V$ in the respectively specified $K$- vector space $V$ is linearly independent a. $K=R$, $ V=R[x] $, $S$= {$x^n-x^m| n,m ∈ R,$ n-even, m-odd}
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1answer
26 views

prove it has basis property

Determine the dimension of the following $K$-vector space $V$, by specifying each having a basis and proving they have Basis property. $K=\mathbb{R}, V= \{ (x_1,x_2,x_3) \in \mathbb{R}^3 \mid ...
1
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1answer
49 views

Proving a strange vector inequality in the euclidean space

It seems to hold the following inequality in an euclidean reference frame $(x,y,z)$: $$\overrightarrow{U}\cdot\overrightarrow{U}\ge\sqrt{2}\left(\Omega_x+\Omega_y\right)$$ where: ...
0
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1answer
33 views

Find subspaces $W$ and $Y$ of $\mathbb{R}^3$ having the property that $W \cup Y$ is not a subspace of $\mathbb{R}^3$.

I'm prepping myself for graduate linear algebra this fall by attempting self-teach myself some of the "basics" of fields, vectors, etc. found in such linear algebra course. I really don't understand ...
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1answer
26 views

Intersection of planes

A line perpendicular to the plane $ 3x-5y+4z-11=0 $ passes through the origin. At what point does this normal intersects the plane?
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1answer
18 views

Minimal polynomial in primary decomposition theorem

I am going over the proof of the primary decomposition theorem. I can prove that if we have an annihilating polynomial $f$ for some linear transformation $T:V \to V$ and $f$ can be expressed $f = ab$ ...
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1answer
16 views

Heading vector from angle (generated by trig) does not have expected result

I am creating a game and when the player taps on the screen, it should generate a ‘pulse’ effect, pushing away the player. For example, the heading vector should have negative x and y values when to ...
0
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1answer
28 views

Help find the equation of two planes

I have the question Consider the line L through the distinct points A = (a,b,c) and D = (d,e,f) Find the equations of the two planes which intersect at right angles along L MY ATTEMPTED SOLUTION I ...
2
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1answer
35 views

Difference between F-space and Frechet space in W. Rudin's “Functional Analysis”

In Walter Rudin's book, "Functional Analysis", we read that by talking about local base, he will be thinking about neighborhoods of $0$. In the vector space context, the term local base will ...
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1answer
19 views

Is there a function to tell if two probability vectors' max values are in the same dimension?

Is there a method or function to tell two probability vectors' max values are in the same dimension? Or Is there a bound for the angle of two normalized probability vector which their max values are ...
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1answer
40 views

Scalar product and Unit vector

Prove that, for any unit vectors $v_1, v_2, \ldots, v_n$ in $\Bbb R^n$, there exists a unit vector $w$ in $\Bbb R^n$ such that $\langle w, v_i \rangle \leq n^{-1/2}$ for all $i=1, 2, \ldots, n$. (Here ...
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1answer
23 views

problem with invariant subspaces

Consider $V$ unitary space. How to prove that if $T:V\rightarrow V$ and $V\ge U$ is invariant subspace of $T$, then $U^⊥$ is invariant subspace of $T^*$. I know the meaning of invariant subspace ...
2
votes
2answers
76 views

Show that no topological vector space is bounded.

I am studying the concept of topological vector spaces in Grubb's Distributions and Operators. A vector space $X$ (over $\mathbb{L} = \mathbb{R}$ or $\mathbb{C}$) is called a topological vector ...
0
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1answer
47 views

Show that S (a subset of V) is contained in span(S)

Let $\text{span}(S) = \lbrace v \in V \mid v\ \text{is a linear combination of vectors in}\ S\rbrace$. I need to show that $S$ is contained within $\text{span}(S)$. I know if $S$ is nonempty, $0$ is ...
0
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0answers
20 views

Reformating Function

Is there such a function where a ambiguous ;n-dimensional, field/space (defined by a function) is plugged in and returns a flattened field where the basic units along the function are then formatted ...
2
votes
1answer
49 views

Linear Functional: Continuous? [duplicate]

Given a Banach space: $E$ and chosen a Hamel basis: $\mathcal{B}$ Any vector induces a (noncanonical) algebraic linear functional by: $$\delta:E\to E^*:\delta_b(b'):=\delta_{b,b'}\text{ defined ...
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1answer
135 views

The intersection of $\operatorname{Im}(T)$ and $\operatorname{Ker}(T)$ is trivial

Let $W$ the space of sequences with entries in $F$ and $S$ the linear operator of $W$ given by: $$S(a_1,a_2,a_3,...) = (a_2,a_3,...)$$ We know that if the intersection of $\operatorname{Im}(T)$ and ...
1
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0answers
18 views

Vectorial product analog operation in 4+ dimensions?

I am thinging about a such operation. Which it need to have: It needs to be $\mathbb{R}^n\times{\mathbb{R}^n}\rightarrow\mathbb{R}^n$ The result needs to be perpendicular to the arguments (thus, ...
2
votes
1answer
33 views

linearly independent vectors and rows/cols space

Given $n$ vectors, we want to determine if those vectors are linearly independent. One way doing it is writing those vectors as columns of a matrix and row-reduce it. The vectors are linearly ...
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0answers
35 views

Geodesic offset of a curve

Are there standard algorithms that can be easily coded to calculate the offset of a geodesic curve? I am working on computer graphics and it is my first time of working on such algorithms. Please ...
2
votes
2answers
57 views

When $ax+by+cz+d=0$ is a plane, $a^2 + b^2 + c^2 \neq 0$

I'm reading a book about equation of planes and an way to determinate the equation is to suppose a point $P = (x, y, z)$ And suppose also that $A=(x_0, y_0, z_0)$ is in the plane. $P$ is in the plane ...
2
votes
3answers
76 views

What is the pushforward of a function (not a vector)

If we have two manifolds $M$, $N$ with the map $f:M \to N$, then this induces a map between their tangent spaces $f_*:T_pM \to T_{f(p)} N$. By duality, another map exists $f^* : T^*_{f(p)}N \to ...
2
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1answer
192 views

Difference between Scalar field and a multivariable Function?

If a scalar field gives out a normal number for every orders pairs what's the difference between it and a function.
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1answer
69 views

Linear Algebra Proof of Injective Function

I'm new in the University and I don't know how to solve this: Suppose $v$ is a non null element of a vector space $V$ on $\mathbb R$. Show that the function is injection: $\mathbb R\to V $ $t ...
0
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2answers
129 views

$\operatorname{span}(S) = V$, finite dimensional. Does there exist a subset of $S$ which is a basis for $V$?

Let $V$ be a finite dimensional vector space and $S \subset V$ a subset (possibly infinite) with $\operatorname{span}(S) = V$. Does there exist a subset of $S$ that is a basis for $V$?
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0answers
21 views

Show that $\{w^{1/2}\phi_n\}$ is an orthonormal set in $L^2(D)$ if $\{\phi_n\}$ is an orthonormal set in $L^2_w(D)$

As mentioned in the title, my problem is: Show that $\{w^{1/2}\phi_n\}$ is an orthonormal set in $L^2(D)$ if $\{\phi_n\}$ is an orthonormal set in $L^2_w(D).$ So I know that: ...
-3
votes
4answers
122 views

Dimensionality and Subspace Existence: A Potential Outlet for Disquisition

The subset of $F^n$ consisting of all vectors $(a_1,a_2,\dots,a_n)$ such that $a_1+a_2+\cdots+a_n=0$ is a subspace of $F^n$ and its dimension is ...(?).... Initially, my intuition said the ...
6
votes
1answer
153 views

Alternative Almost Complex Structures

Let $V$ be a real vector space. An almost complex structure on $V$ is a map $J : V \to V$ such that $J^2 = -\mathrm{id}_V$. An almost complex structure gives $V$ the structure of a complex vector ...
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2answers
120 views

Find tangent vector to surface given a point on the surface and its normal vector (for a sphere)

I need to know how to find a tangent vector to a point on the surface of a sphere if I am given the point P and the normal vector at that point N. I know that there are many possible tangent vectors ...
3
votes
1answer
54 views

Vector spaces - $\min\{p\in\mathbb{N}|\text{ker}f^p=\text{ker}f^{p+1}\}=\min\{q\in\mathbb{N}|\text{im}f^q=\text{im}f^{q+1}\}$

$E$ is a $\mathbb{K}$ vector space, $f\in\mathcal{L}_\mathbb{K}(E)$. Let $p\in\mathbb{N}$ so that $\text{ker} f^p=\text{ker}f^{p+1}$ and $q\in\mathbb{N}$ so that $\text{im} f^q=\text{im}f^{q+1}$ ...
0
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0answers
12 views

Pick a subset of vectors from a set with minimal overlap

This might be more of a comp sci problem, but I was interested more in theory than how to code it. I'm trying to think of a formula/algorithm to do the following: Anyway, so let's say I have a set of ...
0
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1answer
64 views

I want some help of orthogonal vectors

$1.$ Let $u$ and $v$ be orthogonal vectors in $\mathbb{R}^n$ such that $\|u\|=2$ and $\|v\|=3.$ Find $\|2u+3v\|.$ I do it like $\|2u+3v\| < 2\|u\|+3\|v\| = 2.2 + 3.3 = 13$. $2.$ Let $u$ ...
9
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7answers
706 views

Why a subspace of a vector space is useful

I'm in a linear algebra class and am having a hard time wrapping my head around what subspaces of a vector space are useful for (among many other things!). My understanding of a vector space is that, ...
0
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1answer
21 views

Extenstion of Intermediate Value Theorem.

Let $f:[0,1]^{d}\longrightarrow \mathbb{R}^{d}$ with $d\geq 2$. $f$ is continuous and let $c\in (0,1)$. If we have that $f(0,...,0)<<(c,...,c)$ and $f(1,...,1)>>(c,...,c)$, is there an ...
4
votes
1answer
112 views

Bipartite graph matching like problem.

Let $A=\{a_1,a_2, ..., a_n \}$ and $B=\{b_1,...,b_m\}$ be finite sets. Also $A_1,...,A_k\subset A$ are covering of $A$ and $B_1,...,B_t\subset B$ are covering of $B$. Let $V$ be a set of pairs of ...
0
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0answers
33 views

Vector spaces question difficulty

I know how to do ALLL the question parts up to (iv) I just don't know how to show the last part (v) . Please help me.