Take Our Quiz About The Representation On Coordinate Rings

1) What's the other term for an eigenvector?

Eightvector

Vectors

Characteristic vector

Sevenvector

The other term for an eigenvector is a characteristic vector. Eigenvectors are vectors that do not change their direction when a linear transformation is applied to them, only their magnitude is scaled. These vectors are associated with eigenvalues, which represent the scaling factor for the eigenvectors. The term "characteristic vector" is often used in the context of linear algebra to describe these special vectors.

Explanation

The other term for an eigenvector is a characteristic vector. Eigenvectors are vectors that do not change their direction when a linear transformation is applied to them, only their magnitude is scaled. These vectors are associated with eigenvalues, which represent the scaling factor for the eigenvectors. The term "characteristic vector" is often used in the context of linear algebra to describe these special vectors.

2) What's the other term for a polynomial ring?

Polynomial mathematics

Polynomial algebra

Polynomial bond

Polynomials

A polynomial ring is another term for polynomial algebra. This refers to the mathematical structure that deals with polynomials, which are expressions with variables and coefficients. Polynomial algebra involves operations such as addition, subtraction, multiplication, and division of polynomials, as well as the study of their properties and relationships. Therefore, "Polynomial algebra" is the correct answer as it accurately represents the alternative term for a polynomial ring.

Explanation

A polynomial ring is another term for polynomial algebra. This refers to the mathematical structure that deals with polynomials, which are expressions with variables and coefficients. Polynomial algebra involves operations such as addition, subtraction, multiplication, and division of polynomials, as well as the study of their properties and relationships. Therefore, "Polynomial algebra" is the correct answer as it accurately represents the alternative term for a polynomial ring.

3) What's an Affine algebraic variety?

It's the affine n-space K^n

It's the equivalent of K^n

It's simply the n-space

It's the zero-locus in the affine n-space K^n

An affine algebraic variety is defined as the zero-locus in the affine n-space K^n. This means that it is the set of points in the n-dimensional space where a given polynomial equation becomes zero. This definition helps to distinguish an affine algebraic variety from the entire affine n-space and emphasizes its connection to algebraic geometry.

Explanation

An affine algebraic variety is defined as the zero-locus in the affine n-space K^n. This means that it is the set of points in the n-dimensional space where a given polynomial equation becomes zero. This definition helps to distinguish an affine algebraic variety from the entire affine n-space and emphasizes its connection to algebraic geometry.

4) What's a reductive group?

It's a type of linear algebraic group over a field.

It's a type of line of group over a field.

It's a type of linear algebraic group over a curve.

It's a type of continuous algebraic group over a field.

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Explanation

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5) What's a symmetric variety?

It's an algebraic analogue of a symmetric space in general.

It's an algebraic analogue of space in differential geometry.

It's the symmetrical space in differential geometry.

It's an algebraic analogue of a symmetric space in differential geometry.

A symmetric variety is an algebraic analogue of a symmetric space in differential geometry. In differential geometry, a symmetric space is a manifold that is equipped with a symmetric bilinear form. Similarly, in algebraic geometry, a symmetric variety is a variety that possesses a similar symmetry structure. This concept allows for the study of symmetry properties in algebraic settings, providing a bridge between algebraic and differential geometry.

Explanation

A symmetric variety is an algebraic analogue of a symmetric space in differential geometry. In differential geometry, a symmetric space is a manifold that is equipped with a symmetric bilinear form. Similarly, in algebraic geometry, a symmetric variety is a variety that possesses a similar symmetry structure. This concept allows for the study of symmetry properties in algebraic settings, providing a bridge between algebraic and differential geometry.

6) What's a spherical variety?

It's an A-variety with an open dense B-orbit

It's a B-variety with an open dense B-orbit

It's a G-variety with an open dense B-orbit

It's a B-variety with an open dense A-orbit

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Explanation

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7) What's the Peter-Weyl theorem?

It's a basic result in the theory applying to topological groups that are solid but are not necessarily abelian.

The Peter-Weyl theorem is a fundamental result in the theory of harmonic analysis that applies to topological groups. It states that for a topological group that is compact but not necessarily abelian, any square-integrable function on the group can be decomposed into a sum of irreducible representations. This theorem is important in understanding the structure and properties of these groups and has applications in various areas of mathematics and physics.

Explanation

The Peter-Weyl theorem is a fundamental result in the theory of harmonic analysis that applies to topological groups. It states that for a topological group that is compact but not necessarily abelian, any square-integrable function on the group can be decomposed into a sum of irreducible representations. This theorem is important in understanding the structure and properties of these groups and has applications in various areas of mathematics and physics.

8) What's a complex number?

It's a number that can be expressed in the form a-bi

It's a number that can be expressed in the form a+bi

It's a number that can be expressed in the form b+ai

It's a number that can be expressed in the form c+bi

A complex number is a number that can be expressed in the form a+bi, where a and b are real numbers and i is the imaginary unit. The imaginary unit i is defined as the square root of -1. The real part of the complex number is represented by a, while the imaginary part is represented by bi. The complex number system extends the real number system by introducing the concept of imaginary numbers, allowing for the representation of numbers that cannot be expressed as real numbers.

Explanation

A complex number is a number that can be expressed in the form a+bi, where a and b are real numbers and i is the imaginary unit. The imaginary unit i is defined as the square root of -1. The real part of the complex number is represented by a, while the imaginary part is represented by bi. The complex number system extends the real number system by introducing the concept of imaginary numbers, allowing for the representation of numbers that cannot be expressed as real numbers.

9) What's a matrix coefficient?

It's a function on a group of a special form, which depends on a representation of some additional data.

It's a function on a group, which depends on a representation of the group and additional data.

It's a group of a special form, which depends on a linear representation of the group and additional data.

A matrix coefficient is a function on a group of a special form, which depends on a linear representation of the group and additional data. This means that the function is defined on a specific kind of group and is related to a linear representation of that group. The additional data could refer to any extra information that is required to define the function.

Explanation

A matrix coefficient is a function on a group of a special form, which depends on a linear representation of the group and additional data. This means that the function is defined on a specific kind of group and is related to a linear representation of that group. The additional data could refer to any extra information that is required to define the function.