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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2answers
9 views

Given a vector and angle find new vector

Given a vector and an angle, how can i find an vector that the angle between the two vector is exactly the given angle? Edit: We are in the n-dimensional space and the new vector has a fixed given ...
0
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0answers
7 views

How to find Tangent Normal Binornmal vectors of parametric knot

I am given parametric equations of torus knot: $$x = (a+b\cos(qt))\cos(pt)$$ $$y= (a+b\cos(qt))\sin(pt)$$ $$z= c\sin(qt)$$ where $0<t<2\pi$. I need to find Tangent, Normal and Binormal ...
0
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1answer
23 views

Looking for the basis of the kernel of T

Let P$_2$ denote the vector space of all polynomials with real coefficients and of degree at most 2. Define a function T : $P_2$ → $P_2$ by $$ T(P(x)) = x^2 \frac{d^2}{dx^2}(p(x-1))+ ...
1
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0answers
29 views
1
vote
1answer
25 views

For any linear operator $\phi$ on $V$, prove such an integer $m$ exists.

Suppose $V$ is an $n$-dimensional vector space over some infinite number field $K$, $\phi\in\mathcal L(V)$, prove there exists such a (positive) integer $m$ that $$\text{Im} \phi^m=\text{Im} ...
0
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0answers
30 views

What is a change of basis and how do i find it?

W is a four dimensional vector space over a field F with basis S = (v1, v2, v3, v4). B is also a basis of W such that. $b1 =−v1, b2 =v1 +v2, \, b3 =−v1 −v2 −v3, \, and \, b4 =v1 +v2 +v3 −v_4.$ ...
5
votes
1answer
19 views

How to show the sum of the images of such $m$ projections is direct and is the whole space?

There are $m$ projections (whose square are themselves) $\phi_1,\cdots,\phi_m$ acting on a finite-dimensional vector space $V$ such that $$\phi_i\phi_j=0\quad i\ne j\tag{1}$$ where $0$ denotes the ...
3
votes
2answers
47 views

Does the set of all piecewise constant functions form a subspace of the vector space $\mathbb{R}^\mathbb{R}$ over $\mathbb{R}$?

A function $f\in \mathbb{R}^\mathbb{R}$ is piecewise constant if and only if it is a constant function $x\to c$ or there exist $a_1<a_2<\cdots<a_n$ and $c_0,...,c_n$ in $\mathbb{R}$ such that ...
1
vote
1answer
22 views

The weak topology on an infinite dimensional linear space is not first-countable

I thought I needed help proving the above statement, but during typing I found a proof. Since I had already written it all down I will post it anyway, maybe in the future someone can benefit from it. ...
2
votes
1answer
25 views

Dimension of a single coordinate point in $\mathbb{R}^2$?

While going through the comments on an interesting topic on MathOverflow, I came a cross a quote: Take three distinct lines in R^2 as U, V, W. All intersections have 0 dimensions. I have only ...
0
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1answer
36 views

Let $V=\{i\in \mathbb{Z}: 0\leq i< 2^n\}$. Define vector addition and scalar multiplication on $V$ to turn it into a vector space over $GF(2)$.

Let $V=\{i\in \mathbb{Z}: 0\leq i< 2^n\}$ for some $n\in \mathbb{N}$. Define vector addition and scalar multiplication on $V$ in such a way as to turn it into a vector space over the field ...
1
vote
1answer
46 views

How to show T $P_2 \, \to \, P_2$ is a linear operator. Find a basis for the kernel T?

Let P$_2$ denote the vector space of all polynomials with real coefficients and of degree at most 2. Define a function T : $P_2$ → $P_2$ by $$ T(P(x)) = x^2 \frac{d^2}{dx^2}(p(x-1))+ ...
0
votes
0answers
19 views

> Find the matrix A for which $[T(p(x))]_B$= for all p(x) $\in$ P2

Hey i'm quite confused with this question please link me so i can understand the theory. The question is. Let P$_2$ denote the vector space of all polynomials with real coefficients and of degree ...
0
votes
0answers
27 views

$|f(x)-f(y)|\leq u(a,b)|x-y|^t$ for all $a\leq x,y \leq b$. For what values of $t$ is $V_t$ a subspace of $\mathbb{R}^\mathbb{R}$?

For any real number $0<t\leq 1$, let $V_t$ be the set of all functions $f\in \mathbb{R}^\mathbb{R}$ satisfying the condition that if $a<b$ in $\mathbb{R}$ then there exists a real number ...
0
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0answers
24 views

If $A: M \to M$ then $M$ is $A$-invariant subspace and $A $ is an endomorphism?

Just straightening out the terminologies here... Given If $A: M \to M$ then $M$, $M$ some subspace of a vector space, is the following statement equivalent: $M$ is a $A$-invariant subspace $A $is ...
0
votes
2answers
18 views

A set of vectors forming the basis

Can: (b) A set of four vectors: {(1,2,3),(2,3,1),(3,1,2),(1,3,2)} form a basis for $\mathbb{R}^3$?
2
votes
2answers
34 views

Is $V = \{(x,y,z)\in \mathbb{R}^3:\ x+y >1 \}$ a subspace?

Prove whether the following subsets of $\mathbb{R}^3$ are subspaces : (a) $$V = \{(x,y,z)\ \in \mathbb{R}^3:\ x+y >1 \ \},$$ I think that this is not a subspace as the zero vector does not ...
2
votes
1answer
17 views

Finding A Spanning Set

How do I find the spanning set for: $$V = \{(2a,b,0)\ :\ a,b \in \mathbb{R} \},$$ where $V$ is a subspace of $\mathbb{R}^3$?
5
votes
0answers
48 views

Square root of differentiation

Let $T=d/dx$ be the differentiation operator on vector space $V=C^{\infty}(\mathbb{R})$, the space of real (complex) valued smooth maps on real line. To what extent, all subvector space ...
1
vote
2answers
15 views

What am I doing wrong in finding the orthogonal projection of a vector onto the subspace V?

Let $V∈ℝ^5$ be the subspace $V=span{(2,0,0,0,1),(0,2,0,3,0)}$ and let $w=(0,0,-4,-1,-1)$. Find the orthogonal projection of $w$ onto V,using exact values in your answer. My Approach Let the ...
0
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0answers
30 views

Largest algebra in a vector space

Let $V$ be a vector space and let $C$ be a subset of $V$ that is closed under a bilinear operator $\langle \cdot, \cdot \rangle: V \times V \rightarrow V$. Let $A \subset V$ be an algebra containing ...
1
vote
1answer
14 views

Given $k$ distinct linear operators, prove such an $\alpha$ exists

I have $k$ distinct linear operators $\{\phi_i\}$ which act on $V$, a vector space on some number field $K$ (in the sense that $\Bbb Q$ is the smallest possible one). Now I have to prove that there ...
0
votes
0answers
11 views

Given the angles of a 3d vector and the length of one of the components find the length of the other two components

The angles of a vector are 118 with the positive x axis, 76 with the positive y axis and 148 with the positive z axis. The y direction component of the vector is 5. How do you find the other two ...
1
vote
0answers
28 views

Checking if this set is a vector sp.

Question: Define a set $V=\{(x,y):x,y\in\Bbb R\}$. For any two elements $u=(u_1,u_2),v=(v_1,v_2)$ in $V$ and $t\in\Bbb R$, addition and scalar multiplication as, ...
1
vote
1answer
33 views

How are vector space dimension and basis related?

How are vector space dimension and basis related? (I am new to these concepts and know little to nothing about linear algebra/advanced calculus.) Thank you in advance.
0
votes
1answer
15 views

Given a set of complex subspaces, find a set of disjoint subspaces such that every original subspace is the span of the union of some subset

Suppose $S$ is a set of subspaces of $\mathbb{C}^{n}$ for some integer $n$. I would like to find a set $T$ of disjoint subspaces (not just pairwise disjoint - is there a clearer word for this?), such ...
4
votes
1answer
279 views

How can this be a vector space?

I found the following statement: "Example of a linear - vector - space: The set $C^{(k)}[a,b]$ of all (real-valued) continuous functions on a finite interval $a ≤ t ≤ b$ with addition and real number ...
-1
votes
5answers
30 views

Find two vectors v1 and v2 such that when added equal (0, 4, 0).

Struggling with this question. Find two vectors $v_1$ and $v_2$ such that when added equal $(0, 4, 0)$. $v_1$ is parallel to $u(-2, 4, -2)$ and $v_2$ is perpendicular to $u$. Not sure how to start.
2
votes
3answers
52 views

Eigenvalues of matrix with all $1$'s. [on hold]

Let $A$ be the $n \times n$ matrix over a field of characteristic 0, all of whose entries are 1. What are the eigenvalues of $A$, counted with their multiplicities?
1
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0answers
31 views

Looking for references on the complexity of computation of a basis transformation matrix

I'm looking for some references on the complexity for the following kind of problem: Given two Basis $(a_1, ... ,a_n)$ and $(b_1, ..., b_n)$ of the $K(x)$-vector space $V$ I want to compute the ...
0
votes
0answers
33 views

Row rank$=$Column rank

This is one of the proofs given on Wikipedia. Let $A$ be an $m \times n$ matrix with entries in the real numbers whose row rank is $r$. Therefore, the dimension of the row space of $A$ is $r$. Let ...
0
votes
0answers
27 views

Looking for a vector space $V$ and $T \in \mathcal L$ such that $ker (T) \cap Im(T)=\{\theta\}$ but $V \ne ker(T)+Im(T)$

I am looking for example of a linear operator $T$ on a vector space $V$ such that $ker (T) \cap Im(T)=\{\theta\}$ but $V \ne ker(T)+Im(T)$ . I know that $V$ cannot be finite dimensional . Please help ...
0
votes
1answer
23 views

to find basis of homomorphism

Compute $Hom(V,W)$ and also determine its dimension over $F$ where $V$ and $W$ are vector spaces over the Field $F$ given that $V=\mathbb R^2, W=\mathbb R^2, F=\mathbb R$ I have done this: ...
21
votes
3answers
477 views

Is every axiom in the definition of a vector space necessary?

Definition: A vector space over a field $K$ consists of a set $V$ and two binary operations $+: V \times V \to V$ and $\cdot: K \times V \to V$ satisfying the following axioms: ...
0
votes
1answer
30 views

Subspaces of an infinite dimensional vector space

It is well known that all the subspaces of a finite dimensional vector space are finite dimensional. But it is not true in the case of infinite dimensional vector spaces. For example in the vector ...
0
votes
2answers
29 views

Let $V=\mathbb{R}^\mathbb{R}$, let $W$ be the subset of $V$ consisting of all monotonically inc or dec functions. Is $W$ subspace of $V$?

Let $V=\mathbb{R}^\mathbb{R}$ and let $W$ be the subset of $V$ consisting of all monotonically-increasing or monotonically-decreasing functions. Is $W$ a subspace of $V$? Any solutions or hints are ...
0
votes
1answer
36 views

Is there a name for operators of the type $A: M \to M$

In some theorem in functional analysis I have noticed that it is important to assume that an operator $A: M \to M$ where $M$ is some set plus conditions, as opposed to $A: M \to N, M \neq N$ Is ...
1
vote
2answers
22 views

$W$ a subset of $\mathbb{R}^5$ consisting of all vectors an odd number of the entries in which are equal to $0$. Is $W$ a subspace of $\mathbb{R}^5$?

Let $W$ be the subset of $\mathbb{R}^5$ consisting of all vectors an odd number of the entries in which are equal to $0$. Is $W$ a subspace of $\mathbb{R}^5$? I'm not sure how to do this. Any ...
0
votes
0answers
6 views

formula to calculate bounding coordinates of an arc in space

I have an arc in space with known 2 endpoints x1,y1 and x2,y2 centrepoint x3,y3 radius r What would be the formula to find the coordinates of a box that fits the limits of the arc.
1
vote
1answer
24 views

What are those variations of norms called?

Let $V$ be a vector space with a function $\|\cdot\|$ on it that satisfies all the axioms of norms except for scalability condition $\|\alpha \mathbf{x}\| = |\alpha| \|\mathbf{x}\|$ replaced with ...
3
votes
1answer
39 views

Characterization of vectors via $\ell_p$ norms

Suppose you are given all $\ell_p$ norms of a vector $v\in \mathbb R^d$. Is it true that the set of all its $\ell_p$ norms $\{\|v\|_{p},p=1,..,\infty\}$ uniquely define the vector $v$ up to ...
0
votes
2answers
56 views

Proof for $V \cong V^{**}$

Theorem: Let $V$ be an vector space. Then the dual space of $V$'s dual space is canonically isomorphic to $V$. I am able to prove that $V$ is a subspace of $V^{**}$, the map ...
0
votes
1answer
17 views

Computing the characteristic polynomial

Consider the following matrix A over the field $F_7$ $$ \left(\begin{array}{rrr} 3 & 4 & 4 \\ 2 & 5 & 2 \\ 1 & 2 & 5 \end{array}\right) . $$ I'm asked to ...
0
votes
2answers
36 views

Show that the full null space of the matrix A and its column space in the plane 2x+2y - z = 0

Show that the full null space of the matrix A = $\begin{bmatrix} 0&1&5\\ 1&0&0 \\ 2&2&10 \end{bmatrix}$ is the line $\lambda$(0.-5,1), $\lambda \in \mathbb R^3$ and its ...
3
votes
0answers
32 views

Do I have the correct mental map for adjoint operators for inner product spaces?

Let $X$, $Y$ be finite dimensional inner product spaces, let $A: X \to Y$ be a linear operator, let $A^*: Y \to X$ be the adjoint operator to the linear operator, defined using $<y, Ax>_Y = ...
0
votes
1answer
28 views

Union of subspace

Q. Say U and W are subspaces of a a finite dimensional vector space V (over the field of real numbers). Let S be the set-theoretical union of U and W. Which of the following statements is true: a) ...
1
vote
2answers
65 views

What does ||u|| mean?

What does $\left\Vert \mathbf{u}\right\Vert$ mean in this equation? How would this equation be performed? I'm extremely terrible in discrete mathematics and a simplistic answer would be ideal. (Don't ...
0
votes
2answers
25 views

Can we relax the triangle inequality for $\| v \|$ = $\|v - v_0 + v_0\|$?

Given some vector $v$ on vector space $X$ with a norm $\| \cdot \|$ Then $\| v \|$ = $\|v - v_0 + v_0\|$ where $v_0$ is some other vector is it legal to then write $\| v - v_0 + v_0 \| = \|v -v_0\| ...
1
vote
1answer
14 views

Linear transformation with matrices in base

Consider the vector space of real $2 x 2$ matrices and take as base $\{{E_{11},E_{12},E_{21},E_{22}}\}$. Where $E_{ij}$ represents the matrix with a $1$ in the $i$-th row and $j$-th column and the ...
5
votes
1answer
51 views
+50

Closed formula for Poincaré series in terms of adjacency matrix.

Let $Q$ be a finite quiver with vertex set $I$. For each $n = 0, 1, 2, \dots,$ let $k^{(n)}Q \subset kQ$ be the $k$-linear span of all paths of length $n$, in particular, we have$$k^{(0)}Q = ...