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
10 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
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1answer
7 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 ...
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0answers
22 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, ...
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1answer
28 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.
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1answer
12 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 ...
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1answer
274 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 ...
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5answers
28 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.
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3answers
50 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?
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0answers
26 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 ...
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0answers
19 views

quadratic form associated with projection operator in Hilbert space

we are in Hilbert space $L^2 $ 1) we are given subspace of dimension $2K$ $$ V=Vect\{ g_k,\bar{g_k},1\le k\le K \}$$ $V$ is a sum of $K$ subspaces of dimension 2 $$ W_k=Vect \{g_k,\bar{g_k} \} $$ now ...
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0answers
32 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 ...
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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
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1answer
22 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: ...
18
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3answers
406 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
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1answer
29 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
27 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
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2answers
20 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
5 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
38 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
55 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 ...
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votes
2answers
33 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
63 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
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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
0answers
33 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 = ...
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0answers
12 views

Clarification on Sequence space

I have a trivial question but which I'm feeling confused. Is the sequence space a finite collection of vectors whose components are infinite or am I misunderstanding the concept?
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1answer
29 views

Endpoints of a 3D line

How to find the coordinate of the endpoints (A and B) of a line on a surface with known surface normal, center coordinate, and length?
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2answers
59 views

Linear Algebra: which of the definition of subspace of a vector space is more correct?

In a test I was asked to give a definition to a subspace to a vector space, I wrote: A subset $V$ is a subspace of $X$ if $0 \in V$ and $\forall u,v \in V, > \exists \thinspace V$ s.t. $ u+v = ...
0
votes
2answers
42 views

Vector space sizes

$T:V \rightarrow V$ where $V$ is a finite dimensional real vector space I can show that $\ker(T) \subseteq \ker(T^2) \subseteq \ker(T^3) \subseteq\cdots $ Prove there exists some $k$ such that ...
1
vote
1answer
20 views

Are scalar/vector fields in multivariable calculus related to fields of vector spaces in linear algebra

In linear algebra, I have learned that vector spaces are defined over fields. I have to admit that I don't have any background in abstract algebra, so my knowledge of fields are limited to $\mathbb R, ...
1
vote
1answer
43 views

If $V$ is a vector space and $U$ & $W$ are subspaces of $V$, such that $U \oplus W = V$! Need help with proofs!

Consider the map $\rho : V \to V$, defined by $\rho(v) = u − w$, where $v = u + w$, $u \in U$, $w \in W$. Show that: i. $\rho$ is well defined and it is linear; ii. $\rho(u) = u$, $\forall u ∈ U$; ...
0
votes
1answer
39 views

A problem from Finite Dimensional vector spaces

Problem : If $ M $ and $N$ are two subspaces of the vector space $V$ such that $\forall v \in V $ , $ v \in M $ or $ v \in $ (or both) . Prove that at least one of the is equal to $ V $ My ...
1
vote
0answers
14 views

What is the mapping of Z-transform?

Recall that given a series $x(k)$, the Z-transform $\mathcal{Z}$ is defined as: $$\mathcal Z(x(k)) = \sum_{k =0}^{\infty} x(k) z^{-k}$$ where $x(k)$ satisfies $|x(k)| \leq M\rho^k$, $M, \rho$ real ...
0
votes
2answers
56 views

A map that's 1-1 but not onto

I've got some confusion about the definition of a 1-1 map. When I searched for "1-1 correspondence" on Wikipedia, I got redirected to the "bijection" page. So I think the two words mean just the ...
1
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2answers
25 views

Use the cross product to find a parallel vector

I'm confused by this exercise here : Using the cross product, for which value(s) of t the vectors w(1,t,-2) and r(-3,1,6) will be parallel I know that if I use the cross product of two vectors, I ...
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votes
1answer
11 views

A plane and the matter of vector crossing order

I have three 3D points $A$, $B$ and $C$ which are defining a plane. If I want to get the equation of the plane, firstly I need its normal vector. Is it matter if I do it with $AB \times AC$ or $AC ...
2
votes
0answers
67 views
+50

When a vector space will be a complete lattice?

Let $E$ a vector space, and let $P$ a strict cone in $E$ (i.e) $P\subset E$ verify: $$ \mathbb{R}^+ P\subset P \\ P+P\subset P\\ P\cap (-P)=\{0\} $$ So we can easily construct a partial order on $E$ ...
3
votes
0answers
37 views

basis for $\mathbb{R}^{\mathbb{N}}:=\left\{f:\mathbb{N}\to\mathbb{R}\right\}$, and its cardinality.

I know that all vector space has a basis. My question is "concrete" example for basis for $\mathbb{R}$-vector space $\mathbb{R}^{\mathbb{N}}:=\left\{f:\mathbb{N}\to\mathbb{R}\right\}$. If I refer ...
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0answers
12 views

Inner Product of Square Matrices

Let K$^{n*n}$ & M$^{n*n}$ be two square matrices, and K$\cdot$M= \begin{matrix} t_{11} & \cdots & t_{1n} \\ \vdots & \ddots & \vdots \\ t_{n1} & \cdots ...
0
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0answers
12 views

Dot Product of Square Matrices & Inner Product

I need some help! Thank you in advance. Let K$^{n*n}$ & M$^{n*n}$ be two square matrices, and K$\cdot$M= \begin{matrix} t_{11} & \cdots & t_{1n} \\ \vdots & \ddots ...
1
vote
1answer
19 views

Proof that the set of integrable real-valued functions is a vector space

From Folland's Real Analaysis: Modern Techniques and Applications: Proposition: Let $(X,\mathcal{M},\mu)$ be a fixed measure space. The set of integrable real-valued functions on $X$ is a real ...
0
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0answers
13 views

Difference between orthogonal projection onto a point and onto a vector.

A trivial question although I'd like some good answers. Are there any mathematical difference? My vector calculus is a bit rusty.
0
votes
1answer
23 views

Finding the Jordan Normal Form for a General Linear Transformation

Hey everyone here's the problem: Let V be a vector space with dim(V)=n For a particular linear transformation,f, we are given that there are two distinct eigenvalues, λ1 and λ2, with corresponding ...
2
votes
1answer
65 views

Can I assume that the dimension of a vector space is always non-negative?

I'm trying to prove that if $V$ is finite-dimensional and $U_1,...,U_m$ are subspaces of $V$, then $\dim(U_1+...+U_m)\le \dim U_1+...+\dim U_m$ through induction. For $m=1$, the inequality is trivial ...
2
votes
2answers
90 views

Is $(\mathbb{Q},+)$ isomorphic to $(\mathbb{Q}^n,+)$?

Is easy to show that $(\mathbb{R},\mathbb{Q},+,\cdot)$ is isomorphic to $(\mathbb{R}^n,\mathbb{Q},+,\cdot)$ as vectorial spaces and then $(\mathbb{R},+)$ isomorphic to $(\mathbb{R}^n,+)$. This result ...