Prove a few theorems about vector spaces using the axioms

Click For Summary
SUMMARY

This discussion focuses on proving theorems related to vector spaces using axioms. The specific theorems to be proven include: (a) if -v = v, then v = 0; (b) (-r)v = -(rv); (c) r(-v) = -(rv); and (d) v - (-w) = v + w. Each theorem utilizes fundamental properties of vector spaces, emphasizing the relationships between vectors and scalars.

PREREQUISITES
  • Understanding of vector space axioms
  • Familiarity with scalar multiplication
  • Knowledge of vector addition properties
  • Basic grasp of mathematical proof techniques
NEXT STEPS
  • Study the properties of vector spaces in detail
  • Learn about scalar multiplication and its implications
  • Explore mathematical proof techniques for vector space theorems
  • Review examples of vector space axioms in linear algebra
USEFUL FOR

Students of linear algebra, mathematicians, and educators looking to deepen their understanding of vector space properties and theorem proofs.

ataraxia
Messages
1
Reaction score
0
Hey guys,

I need to prove a few theorems about vector spaces using the axioms.

a) Prove: if -v = v, then v = 0

b) Prove: (-r)v = -(rv)

c) Prove: r(-v) = -(rv)

d) Prove: v - (-w) = v + w

where r is a scalar and v, w are vectors.
 
Physics news on Phys.org
really, you need to do those yourself.
 

Similar threads

Undergrad Characterizing linear independence in terms of span
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad About the existence of Hamel basis for vector spaces
  • · Replies 32 ·
2
Replies
32
Views
7K
Undergrad The vector to which a dual vector corresponds
  • · Replies 13 ·
Replies
13
Views
5K
Undergrad Question about vector spaces and subsets
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Confused about small detail in rank-nullity theorem
  • · Replies 1 ·
Replies
1
Views
3K
Graduate Confusion about the Moyal-Weyl twist
  • · Replies 4 ·
Replies
4
Views
4K
Undergrad If T is diagonalizable then is restriction operator diagonalizable?
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad About the definition of vector space of infinite dimension
  • · Replies 18 ·
Replies
18
Views
4K
Graduate A problem in multilinear algebra
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad A = norm-preserving linear map (+other conditions) => A = lin isometry
  • · Replies 4 ·
Replies
4
Views
2K