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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1answer
26 views

Understanding the orthogonal complement of a subspace.

This is my definition of orthogonal complement: Given a vector subspace if $\mathbb{R}^n$, its orthogonal complement is the set of all vectors in $\mathbb{R}^n$ that are orthogonal to any ...
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2answers
88 views

maximum and minimum dimension of the space generated by $\{v_1,v_2,v_3,v_4\}$

I'm confused about this problem. I have four vectors $v_1 = (1,1,1,a), v_2 = (1,2,3,a), v_3= (b,1,0,1), v_4 = (0,b,0,0)$ with $a,b$ real numbers. Determine the maximum and minimum dimension of the ...
0
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1answer
33 views

Find the signature of $Q(x_1,\ldots,x_n)= \sum_{i,j=1}^{n} a_ia_jx_ix_j$

In $\mathbb{R}^n$ let $Q(x_1,\ldots,x_n)= \sum_{i,j=1}^{n} a_ia_jx_ix_j$ quadratic form. $a:=(a_1,\ldots,a_n)\neq0$ $\in \mathbb{R}^n$ find the signature of $Q$
2
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0answers
24 views

About an orthogonal complement theorem

Let $W$ be a subspace of $\mathbb{R}^n$. For any vector $x \in \mathbb{R}^n$, there will one unique vector $y \in W$ that fulfils: $$(x-y) \perp w \ \ : \ \ \forall w \in W$$ I have trouble ...
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2answers
33 views

Determining that this basis is linearly independent with a variable

Have the basis $$B = \{ (1,2,0) , (1,1,1) , (1,a,0) , (0,0,a) \}$$ Explain why doesn't this basis have a dimension of $4$. The only way would be, I guess, that it is linearly dependent, ...
0
votes
1answer
31 views

Coordinate vector of a subspace of $\mathbb{M}_{2,2}(\mathbb{R})$

Have $$\left\{ \left( \begin{matrix} x & y \\ y & x + y \end{matrix} \right) : x,y,\in \mathbb{R}\right \}$$ Which is a vector subspace of $\mathbb{M}_{2,2}(\mathbb{R})$. I was asked ...
1
vote
1answer
35 views

Left shift operator $L: l^2 \rightarrow l^2$ on the sequence space $l^2$

$$L: l^2 \rightarrow l^2$$ is defined by $$b = (b_1,b_2,...) \mapsto Lb = (b_2,b_3,...)$$. $(Lb)_n = b_{n+1}$ respectively. How can I determine the adjoint endomorphism $L^*$? Kind regards George
0
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2answers
22 views

restricting invertible maps to get new maps

For V and W as vector spaces, let we define V ⊗ W and suppose T be a invertible linear map from V ⊗ W to itself with special condition, I want to know whether there exist something like restricted ...
0
votes
2answers
102 views

curl of (cross product of two vectors), i know the formula, but not sure how to prove it

$$\text{curl } \left(\textbf{F}\times \textbf{G}\right) = \textbf{F}\text{ div}\textbf{ G}- \textbf{G}\text{ div}\textbf{ F}+ \left(\textbf{G}\cdot \nabla \right)\textbf{F}- \left(\textbf{F}\cdot ...
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vote
4answers
89 views

Geometrically, what is the span of vectors?

Simple question from a calc 3 beginner. Visually I cannot imagine the span of two vectors, what does this necessarily mean? For example my text mentions if two vectors are parallel their span is a ...
1
vote
1answer
34 views

Is $\mathcal{L}^p \subset \mathcal{L}^{p-1} $?

A random variable $X$ is called integrable if $E[X] < \infty$. We say that $X \in \mathcal{L}^1$ if $E[X] < \infty$, and in general $X \in \mathcal{L}^p$ if $E[|X|^p] < \infty$. I know that ...
0
votes
1answer
26 views

Linear independent vectors of nilpotent transformation

$V$ is a vector space. $N$ is a nilpotent transformation $N:V\rightarrow V$ such that $N^k=0$ ($k$ is the lowest). $v \in V$, $v \notin \text{ker}\ N^{k-1}$ (in other words: $N^{k-1}v \ne 0$). Let ...
0
votes
1answer
12 views

Given two sets of vectors, is there a relationship that describes whether one of them is “orthogonal” to another?

We saw this theorem regarding orthogonal vector subspaces: Have $$A = \{a_1,a_2,a_3,...,a_k\}\\ B = \{b_1,b_2,b_3,...,b_r\}$$ Bases of vector subspaces $S$ and $T$ respectively. Then: ...
1
vote
1answer
22 views

What does the $t$ in $(x,y,z)^t$ mean?

Just a question on notation. I have seen a plane defined this way: $$S = \{(x,y,z)^t \in \mathbb{R}^3 \ / \ 2x-3y+z = 0\}$$ See the $t$ superscript on $(x,y,z)$? Well, I am not quite sure what is ...
2
votes
1answer
39 views

2x2 symmetric matrix is a subspace of vector space.

Can you kindly check my proof of the problem and correct if possible. The following $S=\{A\in M_{2,2} | AA^T=A^TA\}$ is a subspace of $V=M_{2,2}$ all real $2\times2$ matrices. My proof: S ...
0
votes
0answers
23 views

q-analog of vector space dimension

I am reading about "quantum dimension" $\dim_q V$ where $V$ is a vector space. In fact, you could write it $[\dim V]_q$ where $\dim V$ is the dimension of the vector space and $[n]_q = ...
0
votes
3answers
19 views

How to find the vector that passes through a point and is perpendicular to another vector.

Let $ \mathbb{a} = i+4j-3k$ and $b = 7i+20j-12k$ be vectors and $A(2,5,-3)$ be a point. I want find the line $l_ 3$ passing through point $A$ wich is perpendicular to both veotors. How should I do ...
0
votes
1answer
14 views

Eigenvector shared by two endomorhisms

I am guessing if the following fact is true: Let be $V$ a finite vector space above a field $K$. Let $f, g$ be two endomorphisms of $V$ with $f g = g f$. We assume that both $f$ and $g$ have got at ...
0
votes
1answer
23 views

Endomorphisms and Invariant Subspaces

I have a question or two regarding the following exercise: Let $\alpha$ be the endomorphism of $\Bbb{Q}^4$ defined by: $$\alpha : \left[\begin{matrix}a \\ b \\ c \\ d \end{matrix}\right] \mapsto ...
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0answers
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
1answer
44 views

Finding loci of possible points satisfying vector simultaneous equations

I recently had an exam and a question came up which I was only partially able to answer. The question was the following: Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be constant vectors in ...
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votes
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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vote
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 ...
0
votes
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 ...
3
votes
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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votes
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 ...
0
votes
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
vote
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
votes
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 ...
0
votes
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?
0
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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 $f = ab$ ...
0
votes
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 ...
2
votes
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
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
votes
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 ...
1
vote
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 ...
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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vote
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 ...
0
votes
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
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
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 ...