Vector Spaces, Polynomials Over Fields

In summary, a vector space being "over the field of complex numbers" means that the scalars used to multiply vectors within that space come from the set of complex numbers. Similarly, a polynomial being "over the field of complex numbers" means that all its coefficients are complex numbers.
  • #1
Seacow1988
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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!
 
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  • #2
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:
 

1. What is a vector space?

A vector space is a mathematical structure that consists of a set of elements, called vectors, and a set of operations, such as addition and scalar multiplication, that can be performed on these vectors. These operations must follow a specific set of rules, known as axioms, in order for the structure to be considered a vector space.

2. How is a vector space different from a field?

While both vector spaces and fields involve mathematical operations on sets of elements, they have different structures and purposes. A vector space is used to represent and analyze vector quantities, while a field is used to represent and analyze scalar quantities. Additionally, a vector space must be defined over a field, meaning that the elements of a vector space are themselves elements of a field.

3. What are some examples of vector spaces?

Some common examples of vector spaces include Euclidean space, which is the set of all ordered n-tuples of real numbers, and function spaces, which consist of functions that can be added and multiplied by scalars. Other examples include the set of all polynomials of a given degree, or the set of all matrices of a given size.

4. How are polynomials represented in a vector space over a field?

In a vector space over a field, polynomials are represented as vectors, with each coefficient of the polynomial being an element of the field. For example, in the vector space of all polynomials of degree 2 or less over the field of real numbers, the polynomial 2x^2 + 3x - 1 would be represented as the vector (2, 3, -1).

5. What is the significance of polynomials over fields in mathematics?

Polynomials over fields have many important applications in mathematics, including in algebra, geometry, and number theory. They are also used in various areas of science, such as physics and engineering, to model real-world phenomena. Additionally, the study of polynomials over fields has led to important developments in abstract algebra and other branches of mathematics.

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