Could a set of n verctors in Rm span all of Rm when n<m?

  • Thread starter Thread starter yooyo
  • Start date Start date
  • Tags Tags
    Set Span
yooyo
Messages
7
Reaction score
0
Could a set of n verctors in Rm span all of Rm when n<m?
any hits? kinda confused with this span thing.:confused:
 
Physics news on Phys.org
R^m is m dimensional real space (it is easy to write down m independent vectors that span).

Just look at the definitions: the dimension is the minimal cardinality of a spanning set.
 
you still have to prove that less than nvectors cannot span R^n.

i.e. you have to prove that the space of n tuples of real numbers has dimension n.

look at my web notes on linear algebra.
 
Unfortunately, Yooyo did not give any indication as to what he had tried and so we have no idea what facts he can use!

Yooyo, back to you! Are you allowed to use the fact that Rn has dimension n or is proving that part of your problem?
 
can you prove one vector cannot spane R^2?
 
here is a quick inductive argument, if you know about quotient spaces.

case 1, there is no linear surjection from R1 to any higher dimensional space.

if there is a linear surjection from Rn to Rm, where n <m, then the composite surjection from Rn to Rm/em = Rm-1 is not injective.

hence there is a lineaer surjection from some subspace Rn-1 to Rm-1, impossible by inductive hypothesis.
 

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 Why is the dual of Z^n again Z^n ?
Replies
13
Views
2K
I Why can't a set of vectors with less than n elements span ℝn?
Replies
21
Views
4K
I Wronskian: null space equals the span of independent functions
Replies
23
Views
1K
MHB Start on Bland Problem 1, Problem Set 4.1: Generating & Cogenerating Modules
Replies
37
Views
5K
Number of vectors needed to Span R^n
Replies
8
Views
4K
MHB Existence of Tensor Products - Keith Conrad - Tensor Products I - Theorem 3.2
Replies
7
Views
3K
Can the Empty Set Span the Zero Subspace? Insights on Spanning Sets
Replies
5
Views
2K
I When did the consensus on the maximum recession speed in cosmology change?
Replies
7
Views
2K
I Number of Binary Operations on a Set with a Special Property
Replies
2
Views
3K
MHB Proving Span of $\mathbb{R}^2$ Using Sets of Vectors
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top