Questions tagged [linear-independence]

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How to prove for $AX=B$ , number of linearly independent columns in B can't exceed A? [duplicate]

I want a way to prove that for matrices $A(n \times m)$, $X(m \times n)$ and $B(n \times n)$ such that $AX=B$ and the number of linearly independent columns $k$ in $A$ with $k<n$, the maximum ...
Soumya Patel's user avatar
-1 votes
1 answer
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Are any 2 linear combinations of a set of linearly dependent vectors linearly dependent?

I am trying to prove whether the following statement is true or not: that if we have some linearly dependent vectors, any 2 linear combinations of these vectors constitutes a linearly dependent set ...
Princess Mia's user avatar
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How do I approach row elimination for sets that consist of 2x2 matrices instead of 2 x 1 vectors?

I was wondering how to approach row reduction in problems where the given set consists of 2x2 matrices instead of 2x1 vectors. For example: given a set $C$, such that $$C=\{\begin{bmatrix}1 & 0 \\...
Sayeedur Rahman 's user avatar
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4 answers
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Linear independence proof for set of functions

How do I prove that this subset of real valued functions $\{x, \sin{(x)}, \sin{(2x)}\}$ is linearly independent. Here is the proof suggest in the book: Consider the relationship $c_1 . 1 + c_2 . \sin{(...
user763322's user avatar
2 votes
2 answers
161 views

Finding an irrational number between two given irrational numbers constructively

Statement: Let $$X=\{(a,b) \in \mathbb{R} \setminus \mathbb{Q} \times \mathbb{R} \setminus \mathbb{Q}:a<b\}$$ There exists a function $f:X \rightarrow \mathbb{R} \setminus \mathbb{Q}$ such that for ...
Mohammad tahmasbi zade's user avatar
3 votes
3 answers
109 views

Linear dependence condition

We know that to prove vectors $v_1,v_2,v_3$ linearly dependent we must find scalers $x_1,x_2,x_3$ not all equal to $0$ such that $x_1v_1+x_2v_2+x_3v_3=0$. But the doubt I have is on the other ...
a_i_r's user avatar
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Proof verification for dimension of the sum of two spaces

Can anyone tell me if my proof is correct or if there is something lacking in my argument? I am not too confident about it. I used some hints to construct my argument but I took a different route to ...
Hiraethy's user avatar
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30 views

represent linear matroid as an affine matroid

There is an exercise in András Frank’s book Connections in Combinatorial Optimization, An easy exercise shows that every affine matroid can be represented as a linear matroid, and vice versa. I ...
Yu Cong's user avatar
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How do I find linearly dependent groups among vectors?

I have learnt that QR decomposition and RREF can be used to find linearly independent vectors, but now I am confronted with a question that is one step further: How do I find linearly dependent GROUPS ...
张亦弛's user avatar
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Why are these vectors linearly independent? [duplicate]

For vectors $\begin{pmatrix} 1 \\\ 1 \\\ \vdots \\\ 1 \end{pmatrix}$, $\begin{pmatrix} x_1 \\\ x_2 \\\ \vdots \\\ x_n \end{pmatrix}$, $\begin{pmatrix} (x_1)^2 \\\ (x_2)^2 \\\ \vdots \\\ (x_n)^2 \...
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Check the linear independency of a set of cycles in a graph?

As I am reading wikipedia and some material: A basis for cycles of a network is a minimal collection of cycles such that any cycle in the network can be written as a sum of cycles in the basis. Some ...
xue's user avatar
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2 answers
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What Things can be Linearly Independent? [closed]

Is linear independence confined to strictly lists of vectors, or does it extend to vector spaces, subspaces, etc? If it isn't just lists of vectors what are the most common things that can be linearly ...
Jon Schneider's user avatar
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1 answer
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prove that if $A^k =0$ then there is $v \in \mathbb{F}^{n}$ such that the group $\{v,vA,vA^2,\dots,vA^{k-1}\}$ is linearly independent in induction

I have a question which I didn't manage to solve. Assume that there is $A \in \mathbb{F}^{nxn}$ niloptent matrix, which means that there is $k \in \mathbb{N}$ such that $A^k = 0$. I need to prove that ...
DaniDin's user avatar
3 votes
1 answer
56 views

Prove a set is linearly independent.

$\phi = \mathbb{V} \rightarrow \mathbb{V}$ is an operator satisfying $\phi^n = 0$ for some $n$ and $\phi^{n-1} \ne 0$ Let $v \in \mathbb{V}$ be a vector s.t. $\phi^{n-1} \ne 0$. Is the set {v, $\phi(v)...
Avgustine's user avatar
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1 answer
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Injectivity of linear combinations of linearly independent invertible operators

Let $\mathbf{H}$ be an infinite dimensional, separable Hilbert space. Moreover, let $\{A_i\}_{i=1}^N$ be a set of linearly independent, invertible operators that act on $\mathbf{H}$. In other words, $...
Doofenshmert's user avatar
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I have confusion understanding why Zero Vector space has dimension zero? [duplicate]

Here's what I understand from Basis. Basis: it's set of linearly independent vectors which can span the vector space. Basis for Zero vector space: case 1: when { 0 } , it's singleton set , since ...
Inception's user avatar
1 vote
2 answers
71 views

Is the set of vectors $(1, 0), (i, 0)$ a linearly independent subset of $\mathbb{C}^2(\mathbb{R})$?

I've encountered a problem with two approaches, leading to contradictory results. Approach 1: Utilizing the definition of linear independence, consider the equation $c_1 \begin{bmatrix} 1 \\ 0 \end{...
Red_RanGer's user avatar
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Independence of $\partial_t f(t,x)$ and $f(t,x)$ at a fixed time slice?

Let $t_0 \in \mathbb{R}$ is fixed $\mathcal{E}\bigl(\mathbb{R} \times \mathbb{R}^n \bigr)$ be the space of smooth real-valued mappings $f(t,x)$ for $(t,x) \in \mathbb{R} \times \mathbb{R}^n$. Now, ...
Keith's user avatar
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1 vote
3 answers
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If we have $n-1$ dependent vectors of size $n$, why if we remove the $i$th element from each, they're still dependent?

I was reading this answer to the problem of showing $\operatorname{rank}(\operatorname{adj}A)\in\{0,1,n\}$. In the first case we see: If $A$ has rank less than or equal to $n - 2$ then any $n - 1$ ...
Mason Rashford's user avatar
2 votes
0 answers
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If a basis for $V$ intersects an $m$-dimensional subspace of $V$ at $m$ basis vectors, will these $m$ vectors form a basis for the subspace?

Sorry, I might be asking a stupid question, but somehow I can't convince myself of the assertion in the title: Let $\{u_1,\ldots,u_n\}$ be a basis for a vector space $(V,F)$ and suppose $S$ is an $m$-...
Boar's user avatar
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Prove that $y_1(x) = y_{2}(x) \int_{}^{x} \frac{W(\tilde{x})}{y_1(x)^2\, d\tilde{x}$ for a non-vanishing Wronksian

Prove for a non-vanishing Wronksian $y_1(x) = y_{2}(x) \int_{}^{x} \frac{W(\tilde{x})}{y_1(x)^2} \, d\tilde{x}$ I start with normal definition of W, to use find what is $\frac{y_1}{y_2}$: $W = ...
atomic-muclei's user avatar
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1 answer
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Linear independence as a way to gauge predictor usefulness.

Background. The multiple linear regression model is of the form $$ Y = \beta_0 + \beta_1X_1 + \cdots + \beta_nX_n + \epsilon $$ where we assume $\epsilon$ is normally distributed with constant ...
Paul Ash's user avatar
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1 answer
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Subtle conditions that imply linear independence

Suppose you have a set of $2 < k < n$ non-zero, distinct vectors in $\mathbb{R}^n$ with non-negative integer coefficients such that the sum of the entries of each vector is equal to the same ...
Alex's user avatar
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1 answer
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How to find a linear combination of vectors that is congruent to $0\pmod 2$

My goal is to find a linear combination of the following vectors such that the result is $(0,0,0,0,0,0,0,0,0,0,0) \pmod 2$. I've been trying to find a calculator to do it, but even Wolfram Alpha says ...
Cotton Headed Ninnymuggins's user avatar
8 votes
1 answer
174 views

Let $f: [0,1]\to\mathbb{R}$ be injective. Does $\sum_{n=1}^{\infty} c_n\left( f(x)\right)^n=0\forall x\in [0,1] \implies c_n=0\forall n\in\mathbb{N}?$

Let $f: [0,1] \to \mathbb{R}$ be injective. Does $ \displaystyle\sum_{n=1}^{\infty} c_n \left( f(x) \right) ^n = 0\ \forall x\in [0,1] \implies c_n = 0\ \forall n\in\mathbb{N}\ ? $ Maybe something ...
Adam Rubinson's user avatar
1 vote
2 answers
67 views

Is the method to prove that a code [n,k,d] punctured at $i$th index gives [n-1, k_i,d_i] code where $k_i\geq k-1$ is correct?

I am self teaching my self error correcting codes from the book Introduction to coding theory by Ron Roth. There is a question that says that suppose a code is given as $[n,k,d]$ where n is the code ...
Anjanyea's user avatar
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1 answer
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Dimension of space spanned by vectors subtracted by a linear combination of them

I've encountered a problem similar to this one: Linearly independent vectors each subtracted by a linear combination of them are linearly dependent if coefficients add up to 1 To restate it for ...
Vadim's user avatar
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1 answer
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Prove each non-trivial solution, $u(x)$ of a normal, second-order linear ordinary differential equation, has only simple zeroes.

Prove each non-trivial solution, $u(x)$ of a normal, second-order linear ordinary differential equation, $$a_2(x)y'' + a_1(x)y' + a_0(x)y = 0$$ $\forall\ x \in I$ has only simple zeroes. Where point $...
number8's user avatar
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1 vote
0 answers
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Explain why the system of matrices $I, A, A^2, \ldots, A^{n^2 + 1}$ is dependent

For context, one of the homework problems given in my linear algebra classes is this: Given $0 \neq A \in M_n(\mathbb{C})$, prove there is a $k \in \mathbb{N}$ and a monic polynomial $m_A$ of degree $...
Mailbox's user avatar
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2 votes
0 answers
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Can you transpose a matrix before doing rref?

I know that to find whether vectors are independent, you make them the columns of a vector and rref. For example, (1, 2, 4, 5) (2, 3, 4, 7) and (2, 4, 2, 4) become \begin{bmatrix} 1 & 2 & 2 \\\...
kr141's user avatar
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$u_{1},...,u_{n}\in V$ L.I.. Then for any $v_{1},...,v_{n}\in V$, $\exists$ infinitely $\alpha$ s.t. $u_{1}+\alpha v_{1},...,u_{n}+\alpha v_{n}$ L.I..

Let $V$ be a vector space over a field $F$ and $u_{1},...,u_{n}\in V$ are linearly independent. Prove that for any $v_{1},...,v_{n}\in V$, $u_{1}+\alpha v_{1},...,u_{n}+\alpha v_{n}$ are linearly ...
OneLamp's user avatar
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1 answer
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If a vector in a linearly independent list is replaced by a linear combination of itself and one other vector is the result linearly independent?

If one vector in a linearly independent list is replaced by a linear combination of itself and one other vector from that list is the resulting list always linearly independent? It seems to me that ...
NAD's user avatar
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1 vote
3 answers
97 views

How do we prove that $(x^4-6x^3), (x^3-6x^2), (x^2-6x)$, and $(x-6)$ are linearly independent?

Consider the following problem from Axler's Linear Algebra Done Right Let $U=\{p\in P_4(\mathbb{F}): p(6)=0\}$. Find a basis of $U$. $P_4(F)$ is the set of all polynomials of degree $\leq 4$ with ...
xoux's user avatar
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5 votes
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The $n\times n$ matrices $A^n, A^{n-1},\dots,\mathrm{id}_n$ are linearly dependent

Let $A$ be an $n\times n$ matrix over some field $K$. Then its characteristic polynomial $\mathrm{char}_A(X)\in K[X]$ is monic of degree $n$ and annihilated by $A$ (Cayley-Hamilton). It follows that ...
Zuy's user avatar
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1 answer
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Must linearly dependent vectors pass through the origin?

Why it says 'linearly dependent vectors pass through the origin'? Why it must hold? or must it not? Thanks in advance.
Wong's user avatar
  • 143
-1 votes
1 answer
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Linear independence of a matrix with only variables

I am really struggling with the following problem since it has no numbers to find a determinate, I am not sure how to solve/prove this. The question: Let a,b,c in R^3 be linearly independent. Show ...
mirarmar's user avatar
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1 answer
95 views

What are some proofs that the list $1,z,...,z^m$ is linearly independent in $P(\mathbb{F})$ for each nonnegative integer $m$?

In Axler's Linear Algebra Done Right we are asked to verify the assertion that the list $1,z,...,z^m$ is linearly independent (l.i) in $P(\pmb{\mathbb{F}})$ for each nonnegative integer $m$. The way I ...
xoux's user avatar
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Determine if a set of vectors is linearly independent or dependent?

I am "struggling" with a basic question, and would like someone to help me to point out what's wrong in my logic. Let A is a subset of vectors in V(4,3): A = {(0,1,2,1), (1,0,2,2), (1,1,1,0),...
Cooper Brian's user avatar
1 vote
0 answers
44 views

what is the quickest way of finding the set of all linearly independent sets over GF(2) and GF(3)?

Here is the question I am trying to solve (note that in the textbook the columns are numbered from 1 to 6): Let $A$ be the matrix $\begin{pmatrix} 1 & 0 & 0 & 1 & 1 & 0\\ 0 & 1 ...
Emptymind's user avatar
  • 2,049
2 votes
1 answer
94 views

Find invertible matrices over a field such that the set of a vector multiplied by them is linearly independent

Let $\mathbb{K}$ be a field (it may have positive characteristic and/or not be algebraically closed) and let $\mathbb{F} = \overline{\mathbb{K}}$ be it's algebraic closure. Let $0 \neq v \in \mathbb{F}...
Otomeram's user avatar
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1 answer
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Contradiction between RREF & Determinant in evaluating Linear Independence

Working through Anton's Elementary Linear Algebra and there is an example that aims to determine the linear independence or dependence of a system, however I believe the RREF of the coefficients ...
vAlkanol's user avatar
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1 vote
0 answers
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Gram Matrix and Linear Independence on Hilbert spaces source?

I am looking for a quote of the following statement: Vectors $v_1,v_2,...,v_n$ in a Hilbert space $H$ are linearly independent if and only if their gram matrix $G$ defined by $G_{ij}=\langle v_i,v_j \...
Caliondo's user avatar
0 votes
3 answers
76 views

Prove: For full rank matrix $A\in\mathbb{C}^{m\times n}$ with $m>n$ and diagonal matrix $B\in\mathbb{C}^{n\times n}$ we have $rank(ABA^H)=n$

For full rank matrix $A\in\mathbb{C}^{m\times n}$ with $m>n$ and diagonal matrix $B\in\mathbb{C}^{n\times n}$ with $det(B)\ne0$ how can we prove that $rank(ABA^H)=n$? I have already proven that $...
Squid49134's user avatar
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0 answers
61 views

Linear independence of $\begin{pmatrix}1\\ \vdots\\ \frac{1}{n}\end{pmatrix},\cdots,\begin{pmatrix}\frac{1}{n}\\ \vdots\\ \frac{1}{2n-1}\end{pmatrix}$

I want to prove the linear independence of the set of n vectors $$ \begin{pmatrix} 1 \\ \frac{1}{2} \\ \vdots \\ \frac{1}{n}\end{pmatrix}, \begin{pmatrix} \frac{1}{2} \\ \frac{1}{3} \\ \vdots \\ \frac{...
David G's user avatar
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0 answers
138 views

Show that $F(t) = (\cos t, \sin t)$ is locally one-to-one but not globally one-to-one.

I am studying Implicit Function Theorem and Inverse Function Theorem. The problem I want to ask is: Show that $F(t) = (\cos t, \sin t)$ is locally one-to-one but not globally one-to-one. I have two ...
Beerus's user avatar
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1 vote
0 answers
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Prove that set of columns of an incidence matrix M with integral values mod 2 are linearly independent iff the corresponding set of edges are acyclic

This is a question from CLRS 3E, 16-3a. Argue that a set of columns of an incident Matrix M is linearly independent over the field of integers modulo 2 if and only if the corresponding set of edges ...
QuantumStatic's user avatar
2 votes
3 answers
65 views

An analogous equivalence to one of linear independence

In a finite dimensional vector space $V$, a subset $S \subseteq V$ is said to be linearly dependent if there is a nontrivial solution $(a_i)$ to the equation $$ \sum_{\mathbf{v_i}\in S}a_i\mathbf{v_i} ...
Carlyle's user avatar
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0 answers
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Conditions for linear independence given the family of vectors is pairwise independent

let $n>0, n \in \mathbb{N}$ Given $n+1$ vectors $a_0, a_1, ... a_n$ which are pairwise independent i.e. for all $i,j \in \{ 0,1,...,n \}$ and $i\neq j$, $a_i$ and $a_j$ are linearly independent. ...
Darth jedi 's user avatar
2 votes
2 answers
220 views

Generalisation of Stone-Weierstrass Theorem / Fourier series for linearly independent functions.

In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval $[a, b]$ can be uniformly approximated as closely as desired by a ...
Adam Rubinson's user avatar
2 votes
0 answers
42 views

Basis of vector space and linear independence

if a set of n vectors is spanning a n dimensional vector space, then does this means that vectors are linearly independent ? The doubt is coming from the statement which says- " If Dim V =n and ...
Sandeep's user avatar
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