Mathematics

Questions tagged [linear-independence]

177 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…
Elia Immanuel Auer
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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…
Jack
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8
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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
  • 25,208
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 $X$…
Ata
  • 231
6
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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 matrix.…
Caligari
  • 151
5
votes
2 answers

Let $\left \{u_1, u_2, u_3, v_1, v_2, v_3 \right \}$ be vectors in $\mathbb{R}^4$. Let $U$ be the span of $\left \{u_1, u_2, u_3 \right \}$

Let $\left \{u_1, u_2, u_3, v_1, v_2, v_3 \right \}$ be vectors in $\mathbb{R}^4$ . Let $U$ be the span of $\left \{u_1, u_2, u_3 \right \}$ and let $V$ be the span of $\left \{v_1, v_2, v_3 \right \}$ Consider the following statements: (I) If the…
HS077
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5
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$\alpha$ 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
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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 Corollary. $A^n, A^{n-1},\dots,\mathrm{id}_n$ are…
Zuy
  • 5,141
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
  • 17,014
4
votes
2 answers

Compute $\int\left(5\cos^3(t)-6\cos^4(t)+5\cos^5(t)-12\cos^6(t)\right)\,\mathrm{d}t$ using some results in Linear Algebra

This problem appears in the book: Linear Algebra and its applications - David C. Lay - Fourth Edition It appears in: Chapter 4 (Vector Spaces), Section 4.7 (Change of Basis), Exercise 18 $(4.7), \mathbf{[18]:}$ (Calculus Required) Recall from…
Hussain-Alqatari
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4
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4 answers

Find the largest possible number of linearly independent vectors among a given set of vectors

From the book, Linear Algebra by Kuldeep Singh. Miscellaneous Exercise $2,$ Question $2.8:$ Find the largest possible number of linearly independent vectors among $v_1=\begin{bmatrix} 1 \cr -1 \cr 0 \cr 0 \end{bmatrix} v_2= \begin{bmatrix} 1 \cr 0…
Adam Rubinson
  • 25,208
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…
Opsimath
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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
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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
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3
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Finding the degree $d$ of the minimal polynomial of $A$ by finding a vector $X$ such that $X,AX,\ldots,A^{d-1}X$ are linearly independent

Let $A$ be an $n \times n$ matrix over a field $F$ . Let $d(A)$ be the largest non-negative integer such that the matrices $Id, A, A^2,\ldots,A^{d(A)}$ are linearly independent in the vector space of all $n \times n$ matrices over $F$ . Then, $A$…
Avyaktha Achar
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