chatGPT 선생님) 선형대수학 기본 개념이 뭐야

i think you can explain whole key concepts about linear algebra. please write all you know everything as long as possible.

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Linear algebra is the branch of mathematics concerned with the study of linear transformations, or linear maps. It provides a rich theory for describing the behavior of vectors and matrices and the linear transformations that can be applied to them. The main concepts of linear algebra are:

1. Vector spaces: A vector space is a collection of vectors that can be added and multiplied by scalars. The key property of a vector space is that the operations of addition and scalar multiplication satisfy certain axioms.
2. Subspaces: A subspace is a subset of a vector space that is itself a vector space under the same operations. A subspace is typically defined as the span of a set of vectors.
3. Basis and dimension: A basis is a set of linearly independent vectors that span a vector space. The dimension of a vector space is the number of vectors in a basis.
4. Linear transformations: A linear transformation is a mapping between two vector spaces that preserves the operations of addition and scalar multiplication. A linear transformation can be represented by a matrix, and the properties of a linear transformation can be studied by analyzing its associated matrix.
5. Rank and nullity: The rank of a matrix is the number of linearly independent columns in the matrix. The nullity of a matrix is the dimension of its null space, or the space of all vectors that are mapped to the zero vector under the linear transformation defined by the matrix.
6. Isomorphism: Two vector spaces are said to be isomorphic if there exists a bijective linear transformation between them. Isomorphic vector spaces have the same dimension and the same algebraic structure.
7. Orthogonality and orthonormal sets: Orthogonality refers to the concept of perpendicularity between vectors. An orthonormal set is a set of orthogonal vectors that have been normalized to have length 1. Orthonormal sets are often used to form an orthonormal basis for a vector space.
8. Eigenvalues and eigenvectors: An eigenvector of a linear transformation is a non-zero vector that is unchanged by the transformation. The scalar associated with an eigenvector is called its eigenvalue. The eigenvalues and eigenvectors of a linear transformation play a central role in the study of linear algebra, as they provide information about the behavior of the transformation.
9. Determinant and trace: The determinant of a square matrix is a scalar value that describes the scaling factor of the linear transformation defined by the matrix. The trace of a matrix is the sum of its diagonal entries. Both the determinant and the trace can be used to study the properties of a linear transformation.
10. Singular value decomposition (SVD): The singular value decomposition is a factorization of a matrix into a product of matrices that provides important information about the linear transformation defined by the matrix. The SVD is widely used in applications such as image processing, data compression, and machine learning.

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몇 번 시도해본 결과 확실히 디테일하게 물어볼 수록 길게 답변해준다.  어느정도는 말이 되기도 하고, 몇일간에 거쳐 테스트해본 결과 완전히 이상한 말을 한 경우는 없었다. 블로그에 남기는 것 들은 최대한 검증된 내용들로만 채워야겠다.