Scalars and determining subspaces

  • Context: Undergrad 
  • Thread starter Thread starter MoreDrinks
  • Start date Start date
  • Tags Tags
    Scalars Subspaces
Click For Summary
SUMMARY

To determine if a subset W of a vector space V is a subspace, it must be closed under addition and scalar multiplication. This means that for any vectors v and w in W, the sum v + w must also be in W. Additionally, for any scalar α, the product αv must remain in W. These conditions are essential for confirming that W qualifies as a subspace of V.

PREREQUISITES
  • Understanding of vector spaces and their properties
  • Knowledge of scalar multiplication in linear algebra
  • Familiarity with the concepts of closure under addition
  • Basic comprehension of subsets in mathematical contexts
NEXT STEPS
  • Study the properties of vector spaces in linear algebra
  • Learn about the criteria for subspaces in vector spaces
  • Explore examples of subspaces in R^n
  • Investigate the implications of linear combinations in vector spaces
USEFUL FOR

Students of linear algebra, mathematicians, and educators looking to deepen their understanding of vector spaces and subspace criteria.

MoreDrinks
Messages
45
Reaction score
0
To determine if a subset of a vector space is a subspace, it must be closed under addition and scalar multiplication. As far as I can tell, this means adding two arbitrary vectors in the subset and having the sum be within the subset.

But...can the scalar be any number? Is there any limitation?
 
Physics news on Phys.org
If ##W## is a subspace of ##V##, then ##\alpha v\in W## for any scalar ##\alpha## and for any ##w\in W##.
 
micromass said:
If ##W## is a subspace of ##V##, then ##\alpha v\in W## for any scalar ##\alpha## and for any ##w\in W##.

Thanks, my book is terrible.
 

Similar threads

Undergrad Vector subspace and basis vectors in the context of data science
  • · Replies 6 ·
Replies
6
Views
4K
Undergrad Row space, Column space, Null space & Left null space
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad When is a subset a subspace in a vector space?
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Showing that a set of differentiable functions is a subspace of R
  • · Replies 19 ·
Replies
19
Views
6K
MHB Subset that satisfies all but one axioms of subspaces
  • · Replies 16 ·
Replies
16
Views
2K
MHB What Is Required to Prove a Subset is a Vector Space?
  • · Replies 3 ·
Replies
3
Views
3K
Vector Subspaces: Determining U as a Subspace of M4x4 Matrices
  • · Replies 8 ·
Replies
8
Views
2K
MHB Subsets of $\mathbb{R}^2$ Satisfying S2 and S3 but Not S1: Empty Set
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Terminology for Substructures in Vector Spaces
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Nonempty Subspace: Proving 0u = 0
  • · Replies 2 ·
Replies
2
Views
1K