Questions tagged [linear-independence]

Question 1

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 ...

Question 2

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), \...

Question 3

This is an "exercise" from Linear Algebra Done Right. I'm trying to prove the following theorem using the approach outlined by the author: (3.125). Suppose $V$ is a finite-dimensional ...

Question 4

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 $\left \{v_1, v_2, v_3 \right \}$ ...

Question 5

I have a set $S = \{v_1, v_2, \cdots, v_N\}$ with $N$ vectors, each of which have $d$ dimensions. I would now like to find a set $S' = \{n | v_n \in \mathrm{span(}S \setminus \{v_n \}) \}$, i.e. the ...

Question 6

Suppose we have 3 vectors $(a, b, c)$ in the x-y plane and a fourth $(d)$ in the y-z plane. If 4 vectors in 3D space are always linearly dependent, how do we express the fourth in terms of the other 3?...

Question 7

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 \...

Question 8

I've been trying to get my head around this simple exercise question from an old linear algebra exam, but I can't find a good intuition that would help me to solve it. Suppose that $V$ is a vector ...

Question 9

Let's assume we have a piece-wise definition of polynomials, e.g. $p:[0,n]\to\mathbb{R}$ where $n\in\mathbb{N}$ and $$ \begin{align*} p(x):=\begin{cases}p_1(x):=a_1\cdot 1+b_1\cdot x+c_1\cdot x^2,&...

Question 10

Let be $[a,b]$ an interval and $\Delta$ a partition of $[a,b]$ where $a=\xi_0<\xi_1<\dotsc<\xi_n=b$ and $I_j:=(\xi_{j-1},\xi_j)$. Further, let be $l,m\in\mathbb{N}$. We define the space of ...

Question 11

Fix $M, N \in \Bbb{N}$ and define: $$ f_n(x) = \sum_{r = 0}^{MN!}{MN! \choose r} \exp\left(\frac{i2\pi rx}{n}\right), \ n = 1,\ldots,N $$ Conjecture. Then is the set $\{f_n(x), n =1,\ldots,N\}$ a $\...

Question 12

Consider the set $G_k = \{ g_n^k(x) = \cos^{2k}(\frac{2\pi h(x)}{n}) : n = 1..N\}$ where $h(x) \in \Bbb{Z}[x]$ is a polynomial such that for each $k\geq 1$, $G_k$ is a $\Bbb{R}$-linearly independent ...

Question 13

In Peter Szekeres's "A Course in Modern Mathematical Physics", problem 3.8 on page 80 asks: Let $V$ and $W$ be any vector spaces, which are possibly infinite dimensional, and $T : V \to W$ ...

Question 14

Let $m\in\mathbb{N}$ and let $x_i\in \mathbb{R}^3\setminus\{0\}$ for all $i=1,\ldots,m$. Is there any necessary and sufficient condition on $\{x_i:i=1,\ldots,m\}$, for the set $\{x^i \otimes x^i:i,\...

Question 15

This is an exercise from Linear Algebra by Friedberg et al. I couldn't find any related questions, so here it goes. Let $y_1,y_2,\ldots,y_n$ be linearly independent functions in $C^\infty(\mathbb C)$ ...

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Questions tagged [linear-independence]

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

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

Proof of $\dim U^0=\dim V−\dim U$

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 $\left \{u_1, u_2, u_3 \right \}$

Efficiently finding out which vectors from some set are linearly independent of the other vectors in the set

Linear dependency of 4 vectors in 3D space

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

Show that a vector $v_j$ is in the span of $w_1, \dots, w_j$ for each $j = 1, \dots, n$

Find basis/dimension of piecewise defined polynomial vector space

Dimension of polynomial spline space

Conjecture about $\Bbb{C}$-linear independence of functions which are certain sums of the standard exponential functions.

If a finite set of functions $G_k = \{ g_n^k : n = 1..N\}$ is linearly indepdent on a real interval $I$, then so is $\lim_{k \to\infty} G_k$?

Linear independence of subset that maps to linearly independent subset for linear map

Linear Independence of tensor Products of Vectors

Wronskian; null space equals the span of linearly independent functions

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