Set of vectors with each subset forming a basis

  • Thread starter Thread starter Constantinos
  • Start date Start date
  • Tags Tags
    Basis Set Vectors
Constantinos
Messages
80
Reaction score
1
Hey!

Let M and N be two natural numbers and N>M. I want to build a set A with N vectors of size M such that each subset S of A, where |S| = M, contains linearly independent vectors.

Another way to put it is that every S should be a basis for R^M.

Any ideas? Thanks!
 
Physics news on Phys.org
Do you want an explicit construction or a proof that such a set exists?
 
For example, if M= 2, you can take i= <1, 0>, j= <0, 1>, and k= i+ j= <1, 1>. Then any subset of order 2, {i, j}, {i, k}, and {j, k}, is a basis.

For M= 3, start with i= <1, 0, 0>, j=<0, 1, 0>, and k= <0, 0, 1> and add l= i+ j+ k.

Can you continue that?
 
Thread 'Determine whether ##125## is a unit in ##\mathbb{Z_471}##'

Thread 'Determine whether ##125## is a unit in ##\mathbb{Z_471}##'

This is the question, I understand the concept, in ##\mathbb{Z_n}## an element is a is a unit if and only if gcd( a,n) =1. My understanding of backwards substitution, ... i have using Euclidean algorithm, ##471 = 3⋅121 + 108## ##121 = 1⋅108 + 13## ##108 =8⋅13+4## ##13=3⋅4+1## ##4=4⋅1+0## using back-substitution, ##1=13-3⋅4## ##=(121-1⋅108)-3(108-8⋅13)## ... ##= 121-(471-3⋅121)-3⋅471+9⋅121+24⋅121-24(471-3⋅121## ##=121-471+3⋅121-3⋅471+9⋅121+24⋅121-24⋅471+72⋅121##...
View full post »

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 »

Similar threads

I Vector subspace and basis vectors in the context of data science
Replies
6
Views
3K
I Non orthogonal basis and the lines of its coordinate grid
Replies
9
Views
3K
I Standard basis and other bases...
Replies
9
Views
3K
I Confused about small detail in rank-nullity theorem
Replies
1
Views
1K
I S is set of all vectors of form (x,y,z) such that x=y or x =z. Basis?
Replies
5
Views
1K
I Trouble with a passage in Zariski & Samuel's Commutative algebra text
Replies
5
Views
641
I Characterizing linear independence in terms of span
Replies
3
Views
1K
I Change of Basis Matrix vs Transformation matrix in the same basis....
Replies
12
Views
5K
I Showing a set is a basis for a vector space
Replies
2
Views
2K
I Dimension and solution to matrix-vector product
Replies
4
Views
3K

Hot Threads

Recent Insights

Back
Top