Is M2 a Vector Space with Modified Scalar Multiplication?

Click For Summary

Discussion Overview

The discussion centers on whether the set of all 2x2 matrices, denoted as M2, qualifies as a vector space under standard vector addition and a modified form of scalar multiplication. Participants are exploring the implications of this modified scalar multiplication on the axioms that define a vector space.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions if M2 is a vector space with the given definitions and asks for justification if it fails to meet the criteria.
  • Another participant suggests starting with a formal definition of a vector space to clarify the discussion.
  • A further contribution attempts to define a vector space informally, likening it to R^n, but is challenged for lacking a formal definition.
  • There is a call for a more rigorous definition that includes the necessary axioms of a vector space.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the definition of a vector space or the implications of the modified scalar multiplication on M2. Multiple viewpoints on the definition and requirements of a vector space are present.

Contextual Notes

The discussion lacks a formal definition of a vector space as per standard mathematical texts, which may lead to ambiguity in evaluating M2 against the vector space axioms. The modified scalar multiplication's impact on these axioms remains unresolved.

blazelian
Messages
2
Reaction score
0
Is this a vector space?

Let M2 denote the set of all matrices of 2 x 2. Determine if M2 is a vector space when considered with the standard addition of vectors, but with scalar multiplication given by
α*(a b) = (αa b)
(c d) (c αd)
In case M2 fails to be a vector space with these definitions, list at least one axiom that fails to hold. justify you answer.

How do you solve this?
 
Physics news on Phys.org


Start with a good formal definition of a vector space.
 


well a vector space is something tht looks like R^n
 


blazelian said:
well a vector space is something tht looks like R^n
That's not a formal definition. Your text should have a definition of a vector space, including about 10 axioms.
 

Similar threads

Undergrad Prove $$T_{p}M$$ is a vector space with the axioms
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad About the definition of vector space of infinite dimension
  • · Replies 18 ·
Replies
18
Views
4K
Vector Subspaces: Determining U as a Subspace of M4x4 Matrices
  • · Replies 8 ·
Replies
8
Views
2K
Insights Why Vector Spaces Explain The World: A Historical Perspective
  • · Replies 0 ·
Replies
0
Views
9K
High School When are two vectors collinear and coplanar?
  • · Replies 10 ·
Replies
10
Views
3K
Graduate What Topological Vector Spaces have an uncountable Schauder basis?
  • · Replies 6 ·
Replies
6
Views
2K
Determining whether a set is a vector space
  • · Replies 15 ·
Replies
15
Views
3K
Graduate Affine Spaces and Vector Spaces
  • · Replies 82 ·
3
Replies
82
Views
9K
Undergrad Equivalence of alternative definitions of conservative vector fields and line integrals in different metric spaces
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad When is a subset a subspace in a vector space?
  • · Replies 7 ·
Replies
7
Views
3K