# What Do You Know About Schur's Lemma?

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In the representation theory branch of mathematics, Schur's lemma is a popular theorem used for algebras and groups. Despite the fact that the theorem is outright basic, it has been very useful since it was created in the 20th century.
For groups and algebras, the theorem uses different approaches. To find out more about Schur's lemma, sit back and enjoy the following questions.

• 1.

### How do we determine if a ring is strongly indecomposable?

• A.

If its endomorphism ring is a local ring

• B.

If its exomorphism ring is a local ring

• C.

If the external ring is a designer ring

• D.

If it has a well-designed and structured ring

A. If its endomorphism ring is a local ring
Explanation
If a ring's endomorphism ring is a local ring, it means that the only ideals in the endomorphism ring are the zero ideal and the entire ring itself. This implies that there are no nontrivial direct sum decompositions of the ring as a module over itself. Therefore, the ring is strongly indecomposable.

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• 2.

### What is the property of a module of a division ring?

• A.

Decomposable

• B.

Indecomposable

• C.

Compressed state

• D.

Precipitation

B. Indecomposable
Explanation
In the context of a division ring, a module is said to be decomposable if it can be expressed as a direct sum of two non-trivial submodules. On the other hand, a module is called indecomposable if it cannot be decomposed in this way. Therefore, the correct answer for the property of a module of a division ring is indeed "Indecomposable".

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• 3.

### Which one of the following representation theories do we use the theorem for?

• A.

Simultaneous equations

• B.

Jeff theory

• C.

Linear equations

• D.

Algebra

D. Algebra
Explanation
Algebra is the correct answer because representation theory is primarily concerned with studying the ways in which algebraic structures, such as groups, rings, and modules, can be represented by linear transformations of vector spaces. Algebra provides the necessary tools and concepts to analyze and understand these representations and their properties. Simultaneous equations, Jeff theory, and linear equations are all specific topics within algebra that can be studied using representation theory.

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• 4.

### What does the sign φ represent?

• A.

Linear representation

• B.

Geometrical representation

• C.

Non-linear representation

• D.

Planar representation

A. Linear representation
Explanation
The sign φ typically represents the angle between two vectors or the phase angle in complex numbers. In the context of this question, linear representation is the most appropriate choice as it refers to representing data or relationships in a straight line, which aligns with the mathematical meaning of the symbol φ. Geometrical representation, non-linear representation, and planar representation do not specifically relate to the sign φ, making them incorrect options.

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• 5.

### Who discovered the theorem?

• A.

Isaac Shcur

• B.

Issai Schur

• C.

John Schur

• D.

Michael Schur

B. Issai Schur
Explanation
Issai Schur is the correct answer because he was a mathematician who made significant contributions in the field of linear algebra, particularly in the study of matrices and their properties. He is best known for Schur's theorem, which provides a characterization of unitary operators on Hilbert spaces.

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• 6.

### In the theorem, what is the property of V and W?

• A.

Isothermic

• B.

Isomorphic

• C.

Amorphotic

• D.

Geometric

B. Isomorphic
Explanation
The property of V and W in the theorem is isomorphic. Isomorphism refers to a mathematical concept where two mathematical objects have the same structure, preserving the relationships between their elements. In this context, V and W have the same structure and relationships, making them isomorphic.

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• 7.

### In the theorem, what do V and W represent?

• A.

Scalar spaces

• B.

Vector spaces

• C.

Geometric representation

• D.

Concentric spaces

B. Vector spaces
Explanation
The theorem mentioned in the question refers to the representation of vector spaces. In this context, V and W represent the vector spaces being discussed. Vector spaces are mathematical structures that consist of vectors and obey certain rules and properties. They are used to study and analyze various mathematical and physical phenomena, making them a fundamental concept in many fields of science and engineering.

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• 8.

### What happens when V=W?

• A.

Trivial G-linear maps become the identity

• B.

Nontrivial G-linear maps become the identity

• C.

The linear map changes its position

• D.

More linear maps would be formed

B. Nontrivial G-linear maps become the identity
Explanation
When V=W, it means that the vector space V and W are equal. In this case, nontrivial G-linear maps, which are linear maps that are not the identity map, become the identity map. This means that any nontrivial G-linear map that was originally mapping vectors from V to W will now map them to the same vectors in V, essentially becoming the identity map.

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• 9.

### If φ is invertible, what would φ be equal to?

• A.

-1

• B.

0

• C.

1

• D.

0.1

B. 0
Explanation
If φ is invertible, it means that there exists an inverse function φ^(-1). The inverse function undoes the operation of the original function φ. In this case, if φ is equal to 0, then the inverse function φ^(-1) would also be equal to 0. Therefore, if φ is invertible, it would be equal to 0.

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• 10.

### How do we determine that Φ is not bijective?

• A.

Ran(Φ) is larger than N

• B.

Ran(Φ) does not exist

• C.

Ran(Φ) is equal to N

• D.

Ran(Φ) is smaller than N

D. Ran(Φ) is smaller than N
Explanation
The correct answer is "Ran(Φ) is smaller than N". This is because for a function to be bijective, it must be both injective (every element in the domain maps to a unique element in the range) and surjective (every element in the range is mapped to by at least one element in the domain). If the range of Φ (Ran(Φ)) is smaller than the set N, it means that there are elements in N that are not mapped to by any element in the domain, violating the surjectivity condition of a bijective function.

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