Vector Spaces, Polynomials Over Fields

Click For Summary
SUMMARY

A vector space is defined as being "over the field of complex numbers" when the scalars used to multiply the vectors originate from the set of complex numbers. This means that any linear combination of vectors in that space utilizes complex coefficients. Similarly, a polynomial p(x) is considered "over the field of complex numbers" when its coefficients, represented as p(x) = ∑ anxn, are also complex numbers. This establishes a clear relationship between vector spaces and polynomials in the context of complex number fields.

PREREQUISITES
  • Understanding of vector spaces and their properties
  • Familiarity with polynomial functions and their representations
  • Knowledge of fields in abstract algebra, specifically the field of complex numbers
  • Basic linear algebra concepts, including linear combinations
NEXT STEPS
  • Study the properties of vector spaces over different fields, including real and complex numbers
  • Explore polynomial functions and their behavior over various fields
  • Learn about linear transformations and their applications in vector spaces
  • Investigate the implications of field theory in advanced algebra
USEFUL FOR

Students of mathematics, particularly those studying linear algebra and abstract algebra, as well as educators looking to deepen their understanding of vector spaces and polynomials over fields.

Seacow1988
Messages
9
Reaction score
0
Vector Spaces, Polynomials "Over Fields"

What does it mean when a vectors space is "over the field of complex numbers"? Does that mean that scalars used to multiply vectors within that vector space come from the set of complex numbers?

If so, what does it mean when a polynomial, p(x) is "over the field of complex numbers"?

Thanks!
 
Physics news on Phys.org
Welcome to PF!

Hi Seacow1988! Welcome to PF! :smile:
Seacow1988 said:
What does it mean when a vectors space is "over the field of complex numbers"? Does that mean that scalars used to multiply vectors within that vector space come from the set of complex numbers?

Yup! :biggrin:
If so, what does it mean when a polynomial, p(x) is "over the field of complex numbers"?

p(x) = ∑ anxn where all the ans are complex numbers :wink:
 

Similar threads

Undergrad About the definition of vector space of infinite dimension
  • · Replies 18 ·
Replies
18
Views
4K
Undergrad Analogy question for algebraists
  • · Replies 1 ·
Replies
1
Views
2K
Insights Why Vector Spaces Explain The World: A Historical Perspective
  • · Replies 0 ·
Replies
0
Views
9K
Undergrad Quintic Polynomial Tschirnhaus Transform: Possible complex Bring Radicals
  • · Replies 0 ·
Replies
0
Views
3K
High School Can the Diagram in the Article Be Interpreted as Commutative?
  • · Replies 3 ·
Replies
3
Views
2K
Graduate How to visualise complex vector spaces of dimension 2 and above
  • · Replies 4 ·
Replies
4
Views
4K
Undergrad Vector subspace and basis vectors in the context of data science
  • · Replies 6 ·
Replies
6
Views
4K
Undergrad Proving inequality about dimension of quotient vector spaces
  • · Replies 25 ·
Replies
25
Views
4K
Undergrad Row space, Column space, Null space & Left null space
  • · Replies 8 ·
Replies
8
Views
3K
High School Why is it Important that something is a vector space?
  • · Replies 10 ·
Replies
10
Views
5K