Skip to main content

Questions tagged [vector-spaces]

For questions about vector spaces and their properties. More general questions about linear algebra belong under the [linear-algebra] tag. A vector space is a space which consists of elements called "vectors", which can be added and multiplied by scalars

Filter by
Sorted by
Tagged with
-1 votes
1 answer
31 views

Linear Algebra Subspaces problem, MIT

Im trying to figure out how to arrive to the solution for problem number 8 below, Using the method of Problem 6, show that the set $\{a + (a + b)t^2 | a, b ∈ \mathbb R\}$ is a subspace of $P_2$. ...
Kcharliee's user avatar
0 votes
1 answer
34 views

Discriminant of a matrix with respect to change of basis

According to "Commutative Ring Theory" by Matsumura:" If $A$ is a finite $k$-algebra, the trace of an element $\alpha$ of $A$ denoted by $tr_{A/k}(\alpha)$ is the trace of the $k$-...
user631697's user avatar
3 votes
0 answers
50 views

Critique my understanding of the determinant's relation to linear independence

If $det(A) = 0$ then the columns of $A$ are linearly dependent. This is a result that I could recite but struggled to reconcile after having completed my first course in elementary linear algebra. ...
MattKuehr's user avatar
  • 163
0 votes
0 answers
13 views

For the linear operator $T$ on the vector space $V$, find an ordered basis for the $T$-cyclic subspace generated by the vector $z$.

For the linear operator $T$ on the vector space $V$, find an ordered basis for the $T$-cyclic subspace generated by the vector $z$. $$V=M_{2\times 2}(\Bbb R), T(A)=\begin{pmatrix} 1 && 1\\ 2&...
Thomas Finley's user avatar
1 vote
1 answer
25 views

Given an inconsistent overdeterminate system AX=b where $A\in M_{m×n}(R)$ and $b\in R^m$ with rank A=n. Find the least square approx. solution of AX=b

Suppose $A$ is a real matrix of order $m\times n$ with $m>n,b\in\Bbb R^m$ be such that the over determined system of linear equations $AX=b$ is inconsistent and $\text{rank} (A)= n.$ Let $W$ be the ...
Thomas Finley's user avatar
-5 votes
0 answers
33 views

For all $a\in A$ and $c\in C$, there exists $a',c'\in (a,c)$ such that $a'\in A$ and $c'\in C$. What can we conclude about $A,C$? [closed]

The $\mathbb R^n$ space is partitioned into three sets $A,B,C$. Given: $B$ is convex. $A,C$ are non empty. For all open segment $(a,c)$ where $a\in A$ and $c\in C$, there exists $a',c'\in (a,c)$ such ...
High GPA's user avatar
  • 3,768
-3 votes
0 answers
43 views

If there is a point in between that is also in set $X$, then $X$ is open. [closed]

Let $X$ and $Y$ be half spaces. Condition 1: For all $x\in X$ and $y\in Y$, there are points $x'$ and $y'$ between $xy$ such that $x'\in X$ and $y'\in Y$. Can condition 1 implies openness of $X$ ...
High GPA's user avatar
  • 3,768
1 vote
1 answer
26 views

Sub division rings of dimension 2 of division rings

Suppose $A$ is a division ring and $B$ is a sub division ring such that $A$, as a left vector space over $B$, has dimension $2$. Is it true that $B$ must be commutative ?
Boccherini's user avatar
1 vote
1 answer
39 views

Proof that two matrices are row-equivalent iff they have the same nullspace

The matrices are both of size m x n over some field F, obviously. The first direction of this proposition is clear enough, however the opposite direction (same nullspace -> row-equivalence) is ...
Blabla's user avatar
  • 179
0 votes
2 answers
46 views

$\lambda \in \mathbb{C}$ is an eigenvalue of the operator $A$, then $\text{Re}(\lambda) = 0$ AND $H$ is a complex vector space, then $A = iB$.

Let $H$ be a Hilbert space and $A \in B(H)$ such that $A^* = -A$. Prove the following statements: (a) If $H$ is a real vector space, then $\langle Ax, x \rangle = 0$ for every $x \in H$. (b) If $\...
hd1's user avatar
  • 79
0 votes
0 answers
35 views

About strictly convex norm

Let X be a normed vector space. $\| . \|$ is a norm. we said this norm is a strictly convex norm if $$ \forall x,y \in X : \| x\| < 1, \|y\| < 1 \Rightarrow \| \frac{x+y}{2} \| <1 $$ I ...
A12345's user avatar
  • 131
1 vote
1 answer
101 views

A trivial problem about two identical hyperplanes

I've been studying Support Vector Machines lately and encountered a trivial problem about hyperplanes. I use an ordered pair $[w;b]$, where $w\in {\bf R}^n-\{\bf 0\}$ and $b\in \bf R$, to implicitly ...
Jooot's user avatar
  • 15
0 votes
1 answer
39 views

Finding basis for column space of a matrix using only its REF? [closed]

Given the REF for the matrix A = $$\begin{pmatrix}1 & -3 & 0 & 0 \\\ 0 & 1 & 0 & 0 \\\ 0 & 0 & 0 & 0 \\\ 0 & 0 & 0 & 0 \end{pmatrix}$$ can we deduce a ...
Hamza Hamdi's user avatar
1 vote
0 answers
41 views

Dimension of division ring over a sub division ring

Let $L$ be a division ring ("skew field") and $K$ a sub division ring. Now suppose that $L$, as a left vector space over $K$, has finite dimension $m$. Does $L$, as a right vector space over ...
Boccherini's user avatar
1 vote
1 answer
41 views

Geodesic tangent space is a vector space?

I have very little knowledge of Differential Geometry and I'm stuck while reading about General Relativity. Consider defining something called a null geodesic tangent space, in analogy with the ...
SCh's user avatar
  • 200
1 vote
0 answers
18 views

Helmholtz - Hodge decomposition in H(curl)

I'm looking for Helmholtz-Hodge type decompositions but for vector fields slighty more regular than $L^2(D,\mathbb{R}^3)$. I'm familiar with the results in the books of Lions and was wondering if ...
Caillou's user avatar
  • 11
0 votes
0 answers
25 views

Orthonormal basis for complex vector space

I have to answer the question with true or false: Every complex vector space with an inner dot product has an orthonormal basis. I think it is false for the case $\dim V= \infty$. But i cant find a ...
wertz1212's user avatar
1 vote
1 answer
56 views

prove that $\|f\|=1$, where $f : X\to X/\mathscr M$ is the projection

We define $$B(X,Y)= \{ f \mid f:X \to Y, \text{ is continuous and linear function}\}$$ Let $X$ be a normed vector space and let $\mathscr M \subset X $ be a closed subspace. Define $$X/\mathscr M :=\{...
A12345's user avatar
  • 131
2 votes
1 answer
45 views

Defining $\mathcal{A}(S) = TS$ implies $\mathcal{A}$ and $T$ share the same eigenvalues. Need help with proof step.

The following exercise comes from Linear Algebra Done Right, 4th Edition, Sheldon Axler in Section 5A, exercise number 37. Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V)$. Define $\...
Paul Ash's user avatar
  • 1,350
2 votes
1 answer
46 views

About surjectivity of $V\cong (V^*)^*$

Let $V$ be a finite dimensional vector space. We know that $V\cong V^{**}$ naturally . For example, See $V$ is isomorphic to $V^{\ast\ast}$, the double dual space of $V$. My my question is that how ...
Mahtab's user avatar
  • 735
1 vote
1 answer
88 views

Prove that $\|f\|=n^{1/2} $

We define $$B(X,Y)= \{ f \mid f:X \to Y, \text{ is continuous and linear function}\}$$ We said that $f$ is linear bounded function if $$ \exists M>0 : \forall x\in X, \|f(x)\| \le M\|x\| $$ Also, ...
A12345's user avatar
  • 131
-2 votes
0 answers
43 views

Check whether W is a subspace of V. [closed]

Consider the vector space $V = \Bbb R^3$. Consider the subset $W \subseteq V$ containing all vectors $u = (x, y, z)$ such that $x + y + z = 0$. Check whether $W$ is a subspace of $V$.
North-east India's user avatar
1 vote
2 answers
26 views

bv space is a direct sum of $bv_0$ with a one dimensional subspace

We define $bv=\{x=\{x_n\} | \sum_{k=1}^\infty |x_{k+1} -x_k| < \infty , x_k \in \mathbb C\} $ $C_0=\{x=\{x_n\} | lim x_n = 0,x_k \in \mathbb C\} $ And $bv_0=bv \cap c_0$ I need to prove that bv ...
A12345's user avatar
  • 131
0 votes
0 answers
61 views

Solving system of equation [closed]

I have five homogenous equations of six variables: $$ \begin{array}{rcl} a+2b+4c+3d+2e+3f & = &0,\\ 2a+c+3d+3e+f & = & 0\\ a+b+2c+3d+3e+f & = &0,\\ a+c+d+2e+3f & = &0,\\...
David's user avatar
  • 552
0 votes
1 answer
57 views

True or False: Inner product on $\mathbb{R}^2$ satisfying a specific norm.

Verify or refute: There exists an inner product in $\mathbb{R}^2$ such that the norm of every vector $v=(v_1,v_2)$ is $\|v\|=|v_1|+|v_2|$. I think this is untrue. So I took $v=(1,0), y=(0,1)$. After ...
user926356's user avatar
  • 1,380
0 votes
0 answers
5 views

Linear algebra question about span [duplicate]

Prove or disprove the following. If S1 and S2 are arbitrary subsets of a vector space V , then the intersection of their spans (⟨S1⟩ ∩ ⟨S2⟩) equals the span of their intersection (⟨S1 ∩ S2⟩) .
muraleetharan kugaram's user avatar
1 vote
1 answer
59 views

Orthogonal projection is bounded

Definition: Let $U$ be a subspace of $V$. The orthogonal projection of $V$ onto $U$ is the operator $P_U\in L(V)$ given by $$P_U(u+w)=u$$ if $u\in U, w\in U^{\perp}$. Let $V$ be a space with inner ...
user926356's user avatar
  • 1,380
0 votes
1 answer
26 views

Outward pointing normal Tetrahedron

For this tetrahedron I need to write down the order of the vertices such that the normal vector points out of the tetrahedron. For the base DAC, I have drawn the normal vector pointing outwards ...
Dam's user avatar
  • 261
1 vote
1 answer
51 views

What is so special about eigenvector that it behaves like a pivot of linear transformation?

What is so special about eigenvector of matrix $A$ such that matrix $A$ fails to change its direction during linear transformation? The eigenvector (say of matrix $A$) is such a vector in the vector ...
jam's user avatar
  • 63
-1 votes
0 answers
22 views

Decomposition of Vector Space and Linear transformation

I've got stucked in the following problem: Let $V$ be a finite-dimensional vector space over a field $K$. Let $T$ be an endomorphism of $V$. Show that $V$ can be uniquely decomposed as $V_0+V_1$ such ...
Angel Caramón's user avatar
1 vote
1 answer
35 views

Vector space generated by translates

I'm going through a proof in the book Introduction to Banach Spaces: Analysis and Probability and I don't understand some things (I believe that the context is enough, because this is just the ...
Barabara's user avatar
  • 704
-1 votes
0 answers
39 views

a non strictly convex norm on $\mathbb R^2$

Let X be a normed vector space. $\| . \|$ is a norm. we said this norm is a strictly convex if $\forall x,y \in X $ that $ \| x\| \le 1 $ and $ \|y\| \le 1 $ then $ \| \frac{x+y}{2} \| <1$ I need ...
A12345's user avatar
  • 131
0 votes
0 answers
37 views

Defining an equivalence relation through basis elements

Let $V$ be a vectorspace with some basis $\{b_1,b_2,...,\}$. V can be either finite or infinite dimensional. Is defining some basis elements to be in relation, together with definning the relation as ...
16π Cent's user avatar
0 votes
0 answers
24 views

Vector $p$-norm submultiplicativity

Let $X=M_n(\mathbb R)$. Is the $p$-norm on $\mathbb{R}^n$ defined by : $$||x||_p = \left(\sum_{i=1}^n |x_i|^p\right)^{1/p}$$ with $p \in [1,\infty[$ submultiplicative : $||xy||_p \leq ||x||_p||y||_p$ ...
ztg02's user avatar
  • 43
-2 votes
0 answers
25 views

How to find the basis for.a vector space when only its spanning vector is known? [duplicate]

I'm trying to find the basis for a vector space V, which only its spanning vector is known. The problem statement is the following: Find a basis for the vector space V spanned by vectors w1 = (1, 1, ...
Luigi_S_R's user avatar
1 vote
1 answer
25 views

To find the dimension of intersection of two vector spaces.

The question Let $S=\left\{\left(a_1, a_2, a_3, a_4\right): a_i \in \Re, i=1,2,3,4\right.$ and $\left.a_1+a_2+a_3+a_4=0\right\}$ and $\Gamma=\left\{\left(a_1, a_2, a_3, a_4\right): a_i \in \Re, i=1,2,...
amigo's user avatar
  • 39
0 votes
0 answers
54 views

Ambiguity in defining an equivalence relation over a free vector space

In the context of proposing different constructions of tensor, my professor defined the following equivalence relation on the free vector space over $V \times W$ with pointwise addition and pointwise ...
16π Cent's user avatar
0 votes
1 answer
87 views

Can the $\ker T$ and $\ker T\circ T$ be the same?

Let A be an $n\times n$ matrix with entries in $\mathbb{R}$ such that $A$ and $A^2$ have same rank. Consider the linear transformation $T:\mathbb{R^n}\to \mathbb{R^n}$ defined by $T(V)=AV, \forall V\...
math student's user avatar
  • 1,245
0 votes
0 answers
35 views

Subspaces dimension of two 4D planes intersection

Consider vector $\mathbf{v} = (v_1, v_2, v_3, v_4)$. Let $*$ is operation which convert vector $v$ to the vector $\mathbf{v}^* = \left(\dfrac{1}{v_1}, \dfrac{1}{v_2}, \dfrac{1}{v_3}, \dfrac{1}{v_4} \...
Thuja's user avatar
  • 3
2 votes
2 answers
61 views

Why $0$ must be an eigenvalue of any Laplacian matrix?

I asked my teacher one question: Why $0$ be an eigenvalue of any Laplacian matrix, $L$? He tells me below the text: Since the sum of entries along a row/column of $L$ is $0$, $\mathrm{rank}(L)\leq n$...
David's user avatar
  • 552
1 vote
1 answer
77 views

Why eigenspace of -1 orthogonal complement of 1?

Suppose eigenvector of $A(K_n)$ that is orthogonal to $1$ satisfies $A(K_n)v=-1v.$ Hence, $-1$ is an eigenvalue of $K_n$. The argument used to flush out $-1$ revealed also that the eigenspace of $-1$ ...
David's user avatar
  • 552
2 votes
1 answer
48 views

Consequences of definition of scalar product

Definition: Let $V$ be a vector space over the field $K=\mathbb{R}$ (or over $K=\mathbb{C})$. The scalar product on $V$ is a function $V\times V\to K,$ denoted by $(x,y)\mapsto \langle x,y\rangle$, ...
user926356's user avatar
  • 1,380
0 votes
1 answer
40 views

Are all matrices equipped by definition with a matrix-vector product?

I am trying to be as precise as possible with the definitions for a numerical mathematics code that I am developing. This is a very synthetic summary of what I understand so far: I understand that a <...
Nicolás Valle Marchante's user avatar
0 votes
1 answer
69 views

Dimension of a linear subspace of $2^S$ spanned by $100$ singletons?

Consider some set $\mathbf{S}$ containing $2024$ elements. On $2^\mathbf{S}$, define two operations, the addition and the multiplication by elements of $\mathbb{Z}_2$, as $$ A + B = (A \setminus B) \...
Thuja's user avatar
  • 3
0 votes
0 answers
14 views

Find a generating subset of a set of binary vectors such that the number of even sums of generators in the set is minimised.

More precisely: Let $S\in \mathbb{F}_2^n$ be a set of $n$-dimensional binary vectors, find a generating subset $G\subseteq S$ such that the number of elements of $S$ expressible as a sum of an even ...
DeafIdiotGod's user avatar
0 votes
0 answers
10 views

Distance Between Space and Time-Dependent Functions in the $2$-Norm

Suppose I want to measure the "distance" between two functions $u$ and $u_h: [0,T] \times (\Omega\subset\mathbb{R}^2) \to \mathbb{R}$. How would I go about this? I know that, for example, if ...
zaccandels's user avatar
1 vote
1 answer
53 views

How can we be sure that the triangle inequality works in this case?

Consider two complex numbers $z$ and $w$ such that $|z| = |w| = 3$ and $|z - w| = 3\sqrt{2}.$ What is the minimum value of $P = |z + 1 + i| + |w - 2 + 5i|$? This is a question from our country's ...
ten_to_tenth's user avatar
  • 1,078
0 votes
2 answers
42 views

Writing convention of Courant–Fischer theorem

Let $A \in \mathcal{M}_n(\mathbb{R})$ be a symmetric matrix and $\lambda_1 \leq \lambda_2\dots\leq\lambda_n$ be its real eigenvalues taken with multiplicities. Let $1\leq i_1 \leq i_2\dots\leq i_k\leq ...
David's user avatar
  • 552
1 vote
2 answers
86 views

Why $X^\perp=\text{span}\space\{x_{k+1}\dots,x_n\}$ true?

Given the set $X=\{x_1,x_2,\dots\,x_k\}$ be a orthonormal set of eigenvectors of a symmetric matrix $A\in\mathcal M_n(\mathbb{R})$. Then I don't understand why $X^\perp=\text{span}\space\{x_{k+1}\dots,...
David's user avatar
  • 552
1 vote
1 answer
73 views

Different versions of Courant–Fischer theorem

Let $A \in \mathcal{M}_n(\mathbb{R})$ be a symmetric matrix and $\lambda_1 \leq \lambda_2\dots\leq\lambda_n$ be its real eigenvalues taken with multiplicities. Let $1\leq i_1 \leq i_2\dots\leq i_k\leq ...
David's user avatar
  • 552

1
2 3 4 5
365