집합 기본 개념 정리 및 관계

먼저 용어 정리부터 

정의역(Domain) : f: X --> Y 인 함수가 있을때 X라고 생각하면 된다. Y에 원소 대응되는 집합

공역(Co-Domain) : 정의역과 짝을 이룬 다는... 함께 간다는 의미로 Co- 가 붙는다고 한다

Null space(영공간, kernel) : 공역의 0벡터로 맵핑되는 도메인의 모든 벡터 집합 

Nullity : Null space의 차원

Range(상공간, image) : 치역...이라고 생각하면 된다. Y집합에서 실제 X의 원소와 대응되는 집합

Rank: Range의 차원

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그래... 그러면 이것들의 관계들이 뭘까..?(그냥 대책없이 전체의 관계가 궁금하네)

In linear algebra, what is mean & relationship like rank, nullity, subspace, null space, zero vector ?

In linear algebra, the concepts of rank, nullity, subspace, null space, and the zero vector are related to the study of linear transformations and matrices.

• The rank of a matrix is the dimension of the column space of the matrix, which is the largest linearly independent set of columns of the matrix. It represents the number of linearly independent columns of a matrix.
• The nullity of a matrix is the dimension of the null space of the matrix, which is the set of all vectors that are mapped to the zero vector by a linear transformation represented by the matrix. It represents the number of linearly dependent columns of a matrix.
• A subspace is a subset of a vector space that is itself a vector space under the same operations. It must contain the zero vector and be closed under addition and scalar multiplication.
• The null space of a matrix is the set of all vectors that are mapped to the zero vector by a linear transformation represented by the matrix.
• The zero vector is a vector in a vector space with all components equal to zero. It is the additive identity element in a vector space, meaning that adding the zero vector to any other vector in the space results in the same vector.

These concepts are important in linear algebra because they help us understand the structure and properties of linear transformations, matrices, and the vector spaces they act on. They are used in many areas of mathematics, science, and engineering, such as computer graphics, control systems, and robotics.

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Is correct that zero vector always exists in any subspace?

Yes, the zero vector always exists in any subspace. A subspace is a subset of a vector space that contains the zero vector and is closed under vector addition and scalar multiplication. This means that if two vectors are in the subspace, their sum is also in the subspace, and if a vector is in the subspace, then any scalar multiple of that vector is also in the subspace. Since the zero vector is the additive identity, it must be in any subspace.

zero vector는 항상 subspace에 존재한다구! 부분 공간은 0 벡터를 포함하면서 +, * 에 대해서 닫힌 벡터 공간이야! 0 벡터는 덧셈의 항등식이기 때문에 모든 부분공간에 있어야해. (행렬식의 곱에서 I 항등식과 비슷한 개념같다. 대각원소들이 전부 1 인거)

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Hmm.. so is there possible when nullity = 0 ?

 

Yes, it is possible for the nullity of a linear transformation to be 0. The nullity of a linear transformation is the dimension of its null space, which is the set of all vectors in the domain that are mapped to the zero vector under the transformation. A linear transformation with a nullity of 0 means that its null space only contains the zero vector, and therefore it is one-to-one (or invertible) and maps every non-zero vector in the domain to a unique non-zero vector in the codomain.

 

Nullity = 0 의 의미는 {0} Null space 안에 zero vector만을 포함하고 있을때 이다.!!

Null space 안에 여러 다른 집합들이 있을 수도 있겠지... 하지만 0 벡터 외에 아무것도 없으면 0인 것이다!

(처음에는 아니 zero vector는 은 항상 있다면서... 그러면 nullity 가 0이면 뭐지..? 생각하다가 선생님이 이해할 수 있게 말씀해주셨다)

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