Questions tagged [linear-independence]
169 questions
12
votes
1 answer
A module with arbitrarily large finite linearly independent subsets but without an infinite linearly independent subset
Does there exist an example of a commutative ring with unity $R$ and an $R$-module $M$ such that:
If $L \subseteq M$ is linearly independent, $|L| < \infty$.
For any $N \in \mathbb{N}$, there exists a linearly independent $L \subseteq M$ with $|L|…
Smiley1000
- 4,219
8
votes
2 answers
Why do I need this extra condition on a vector space basis theorem?
My course notes for an abstract algebra course include this theorem: Let $V$ be a vector space. If $L \subset V$ is a linearly independent subset, and $E$ is minimal amongst all generating sets of $V$ with the property that $L \subseteq E$, then $E$…
Jack
- 804
8
votes
1 answer
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 related to (i.e. a more general version of)…
Adam Rubinson
- 24,300
6
votes
3 answers
Eigenvectors of different eigenvalues are linearly independent (without matrices)
Usually the independence of eigenvalues are shown for matrices, I did a proof without considering matrices.
Let $T:V\to V$ be a linear operator and let $X=${$v_1,v_2,...,v_m$} be eigenvectors each corresponding to a different eigenvalue then $X$ is…
Ata
- 231
6
votes
0 answers
A natural (?) proof of linear indepedence of eigenvectors of distinct eigenvalues.
Proposition. Let $T\colon V \to V$ be a linear operator. If $v_1, v_2, \ldots, v_m$ are eigenvectors of $T$ that belong to distinct eigenvalues, then they are linearly independent.
The usual proofs are (1) by induction, (2) via a Vandermonde…
Caligari
- 131
5
votes
1 answer
$\alpha$ is a column vector with n entries, prove $\{\alpha, A\alpha , A^2\alpha,\cdots, A^{n-1}\alpha\}$ linearly independent
the question is that
Suppose $\alpha $ is a column vector with n entries and $\alpha$ is not zero vector, then there exists square matrix $A$ (n by n) such that {$\alpha, A\alpha , A^2\alpha,\cdots, A^{n-1}\alpha$} are linearly independent.
What I…
UsedT0Be
- 65
5
votes
0 answers
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
Corollary. $A^n, A^{n-1},\dots,\mathrm{id}_n$ are…
Zuy
- 4,962
5
votes
1 answer
When will "permuted vectors" be linearly independent?
Let $n\geq2$ be a natural number and let $x_1,\ldots,x_n$ be $n$ real numbers. Is there a general sufficient condition to guarantee that the set of $n$ "cyclicly permuted" vectors $\left\{(x_1,x_2,\ldots,x_n), (x_n, x_1,\ldots,x_{n-1}), \ldots,…
awllower
- 16,926
4
votes
1 answer
Linearly independent vectors generated from matrices
I came across this question recently:
Do there exist $n$ real $n\times n$ matrices $A_1, A_2, \cdots, A_n$, such that for any
$n$-dimensional non-zero vector $v$, $A_1 v, A_2 v, \cdots, A_n v$ are always linearly independent?
It seems like a usual…
Opsimath
- 41
4
votes
0 answers
How to prove that the functions cot(x), cot(2x), ..., cot(nx) are linearly independent?
I'm trying to prove that the functions:
$cot(x), cot(2x), \dots, cot(nx)$ are linearly independent.
My idea was to use mathematical induction, that is:
For $n = 1, \hspace{0.5cm} \alpha_1 cot(x) \equiv 0 \Leftrightarrow \alpha_1 = 0 $
Now suppose…
John
- 332
4
votes
3 answers
Linear independence in $V =$ the space of functions from $\mathbb R$ to $\mathbb R$
Problem 8.25 in Schaum's 3000 Solved Problems in Linear Algebra states:
Let $V$ be the vector space of functions from $\mathbb R$ into $\mathbb R$. Show that $\{ e^{2t}, t^2, t\} $ is a set of linearly independent vectors.
The answer is supplied…
Hank Igoe
- 1,450
3
votes
3 answers
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 alternative condition,which says if one of them can be…
a_i_r
- 739
3
votes
1 answer
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, $\phi(v)$, $\dots$, $\phi^{n-1}(v)$} linearly…
Avgustine
- 317
3
votes
2 answers
Linear Algebra Done Right confusing proof
In this proof author tries to prove $m \leq n$ but then assumes he can do this process m times. How can we be sure about we don't run out of n?
If this is not the case, why this proof is too long? Can't we just use the Linear Dependence Lemma to…
Ali
- 45
3
votes
2 answers
Fast way to check linear independence of matrix
Say we suspect the columns of a matrix are independent and want to verify that fact quickly by hand. What is the best way to do it?
I'm currently studying MITx 18.033 where they recommend checking if the nullspace is $\{\mathbf 0\}$ by reducing…
Zaz
- 1,544