MHB Linear Dependence of Vectors Spanning a Space: Example Needed

Yankel
Messages
390
Reaction score
0
Hello

A base of some space is a set of vectors which span the space, and are also linearly independent.

I am looking for an example of vectors which DO span some space, but are dependent and thus not a base...can anyone give me a simple example of such a case ?

thanks !
 
Physics news on Phys.org
The easiest way I can think of generalizing this is to start with a basis and then just add one more vector to the set which is a multiple of one of the other vectors.

For example, the vectors [math]\left \{ (1,0),(0,1),(1,2) \right \}[/math] span $\mathbb{R}^2$ but this set isn't a basis.
 
exactly what I was looking for, thanks !
 

Thread 'Trouble understanding an online solution to an exercise in Dummit & Foote'

##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
View full post »

Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
View full post »

Thread 'Decomposition into irreps of compact Lie group'

When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
View full post »

Similar threads

I Characterizing linear independence in terms of span
Replies
3
Views
1K
I Confused about small detail in rank-nullity theorem
Replies
1
Views
1K
I Row space, Column space, Null space & Left null space
Replies
8
Views
3K
I Wronskian: null space equals the span of independent functions
Replies
23
Views
1K
I Proving linear independence of two functions in a vector space
Replies
5
Views
2K
MHB Statements with linearly independent vectors
Replies
10
Views
2K
I Finding the dimension of the span ##u_1-u_2, ...., u_n-u_1##?
Replies
2
Views
3K
B What unique contributions to math did linear algebra make?
Replies
19
Views
3K
I Standard basis and other bases...
Replies
9
Views
3K
I Question about an eigenvalue problem: range space
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top