Can n Vectors in Rm Span All of Rm?

  • Thread starter Thread starter MustangGt94
  • Start date Start date
  • Tags Tags
    Span Vectors
Click For Summary
SUMMARY

A set of n vectors in Rm can span all of Rm if n is greater than or equal to m. This is due to the requirement of having a pivot position in each column of the matrix formed by these vectors. For instance, three vectors in R4 can span R4 if they are linearly independent and provide a pivot in each column. A single vector cannot span R2 as it lacks the necessary dimensionality to cover the entire space.

PREREQUISITES
  • Understanding of linear independence
  • Knowledge of vector spaces
  • Familiarity with pivot positions in matrices
  • Basic concepts of dimensionality in linear algebra
NEXT STEPS
  • Study linear independence and its implications in vector spaces
  • Learn about the rank of a matrix and its relationship to spanning sets
  • Explore the concept of basis in Rm and how it relates to spanning
  • Investigate the role of Gaussian elimination in determining pivot positions
USEFUL FOR

Students of linear algebra, educators teaching vector spaces, and anyone interested in the foundational concepts of spanning sets in mathematics.

MustangGt94
Messages
9
Reaction score
0
Not sure if this is "beyond" Calculus but the prereq for the class was Calc II so I thought i'd post here.

Homework Statement


Could a set of n vectors in Rm span all of Rm?

Homework Equations



None

The Attempt at a Solution


Hmm really stuck at this one. I think that it should be able to span since the only thing that matters is if you have a pivot point in each column? So even if you have say 3 vectors in R4 it could still span it, since you can have still have a pivot point in each of the column if the vectors allow it?

Thanks for the help.
 
Physics news on Phys.org
n<m, right? Can one vector span R2? Why or why not?
 

Similar threads

Given specific v, dimension of subspace of L(V, W) where Tv=0?
  • · Replies 15 ·
Replies
15
Views
3K
Determine which system of vectors span C^3
  • · Replies 19 ·
Replies
19
Views
2K
Show that a set of vectors spans a subspace
  • · Replies 4 ·
Replies
4
Views
4K
Does the Null Space of a 2x3 Matrix Determine its Column Space?
  • · Replies 15 ·
Replies
15
Views
3K
Help with linear algebra: vectorspace and subspace
  • · Replies 15 ·
Replies
15
Views
3K
Finding the Dimension of a Vector Space: The Matrix Method
  • · Replies 2 ·
Replies
2
Views
2K
Linear algebra, can A be one-to-one given a case
  • · Replies 3 ·
Replies
3
Views
1K
Linear Combinations and Span (Concept Question)
  • · Replies 8 ·
Replies
8
Views
2K
Number of indie vectors ##\leq ## cardinality of spanning set
  • · Replies 3 ·
Replies
3
Views
2K
[Linear Algebra] Construct an n x 3 matrix D such that AD=I3
  • · Replies 3 ·
Replies
3
Views
4K