선형 대수학 개론
선형 대수학에 대한 현대적인 소개를 제공하는 포괄적인 교과서로, 벡터, 행렬, 선형 방정식, 벡터 공간, 직교성, 행렬식, 고유값 등을 다루며 이론과 응용에 중점을 둡니다.
강좌 개요
📚 콘텐츠 요약
벡터, 행렬, 선형 방정식, 벡터 공간, 직교성, 행렬식, 고유값 등을 중심으로 현대적 소개를 제공하는 종합적인 교과서이며, 이론과 응용에 모두 중점을 둔다.
기일드 스트랭의 직관적이고 엄격한 프레임워크를 통해 선형 대수학의 기초와 응용을 마스터하세요.
저자: 기일드 스트랭
감사의 말: 매사추세츠 공과대학교; 웨슬리-캔버리지 출판사
🎯 학습 목표
- 선형 조합, 내적, 벡터 길이, 벡터 간 각도를 계산한다.
- 벡터 집합이 형성하는 기하학적 구성(선, 평면, 부피)을 설명한다.
- 행렬-벡터 곱을 사용하여 선형 방정식을 풀고 역행렬과 특이 행렬의 역할을 해석한다.
- 시스템의 행 그림(평면의 교차)과 열 그림(벡터의 선형 조합)을 구분한다.
- 가우스 소거법을 수행하여 시스템을 상삼각형 형태(U)로 변환하고 역대입법으로 해결한다.
- 기본 행렬(E_{ij})과 치환(P_{ij})을 사용하여 소거 단계를 정식화한다.
- 벡터 부분집합이 부분공간이 되는 조건을 만족하는지 확인한다.
- 행소거를 수행하여 기약 행사다각형 형태(R)에 도달하고, 계수, 피봇 열, 자유 변수를 식별한다.
- 특수해로부터 영공간 행렬 N을 구성하고 선형 시스템의 완전한 해를 설명한다.
- 네 가지 기본 부분공간의 직교성을 식별하고 증명하며, 직교 보완을 결정한다.
수업 共 10 课时 · 预计 30.0h
수업
Lesson
This lesson introduces the fundamentals of linear algebra by defining vectors as column components and exploring how scalar multiplication and vector addition create linear combinations. Students learn how these operations form the basis for solving systems of equations and how the concept of span determines the geometric dimensions reachable by a set of vectors.
This lesson explores the dual perspectives of linear systems, contrasting the row-based geometric intersection of planes with the column-based linear combination of vectors. Students will learn to solve $Ax=b$ by interpreting matrices as linear transformations and applying elimination techniques to identify singular matrices and unique solutions.
This lesson introduces vector spaces as collections of objects that satisfy eight fundamental axioms, emphasizing the importance of the zero vector and closure under addition and scalar multiplication. It further explores the column space of a matrix, defining it as the span of its columns and explaining that a system $Ax = b$ is solvable if and only if $b$ lies within that space.
This lesson explores the geometric relationship between the four fundamental subspaces of a matrix, defining them as pairs of orthogonal complements. It further introduces the concept of least squares approximations and projections, explaining how to find the best possible solution to inconsistent systems by minimizing error.
This lesson explores the fundamental properties of determinants, focusing on the product rule, the impact of transposes, and the behavior of orthogonal matrices. It also introduces efficient computation techniques, demonstrating how Gaussian elimination and pivot products simplify the process of finding determinants for complex matrices.
This lesson explores the geometry and algebra of eigenvalues and eigenvectors, explaining how they represent the principal axes of linear transformations where a matrix acts only by scaling. Students will learn to calculate these values using the characteristic equation $\det(A - \lambda I) = 0$ and apply properties like the trace and determinant to analyze matrix behavior and system stability.
This lesson defines linear transformations as mappings that preserve vector addition and scalar multiplication, governed by the principles of additivity and homogeneity. Students learn to identify linear transformations using the "origin test," construct matrices based on how transformations affect basis vectors, and apply these concepts to geometric and algebraic problems.
This lesson explores the pseudoinverse $A^+$ as a powerful tool for solving singular or overdetermined systems, acting as a bridge between the four fundamental subspaces. It further examines the $A^TCA$ framework, demonstrating how this structure models physical equilibrium and statistical least squares by projecting vectors onto the column and row spaces of a matrix.
This lesson explores how Numerical Linear Algebra optimizes computational performance by using homogeneous coordinates to convert affine translations into matrix-vector multiplications. By unifying these operations, developers can leverage hardware-accelerated BLAS routines to process complex geometric transformations with maximum efficiency and structural integrity.
This lesson introduces complex numbers as a foundational extension of the real number system, defined by the identity $i^2 = -1$. By visualizing these numbers as vectors in the complex plane, students learn to perform algebraic operations and interpret complex values as tools for modeling physical phenomena like oscillation and rotation.