선형대수학개론 렉쳐노트 Introduction to Linear Algebra Lecture Notes

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Howard Anton, Robert C. Busby - Contemporary Linear Algebra-John Wiley & Sons (2002) 도서를 골자로 한 선형대수학개론 A+ 렉처노트입니다.

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목차

2.1. Introduction to Systems of Linear Equations
2.2. Solving Linear Systems by Row Reduction

3.3 Elementary Matrices; A Method for Finding A -1
3.4 Subspaces and Linear Independence
3.5 The Geometry of Linear Systems
3.6 Matrices with Special Forms
3.7 Matrix Factorizations; LU-Decomposition

4.1 Determinants; Cofactor Expansion
4.2 Properties of Determinants
4.3 Cramer's Rule; Formula for A-1; Applications of Determinants
4.4 A First Look at Eigenvalues and Eigenvectors

6.1 Matrices as Transformations
6.2 Geometry of Linear Operators
6.3 Kernel and Range
6.4 Composition and Invertibility of Linear Transformations

7.1 Basis and Dimension
7.2 Properties of Bases
7.3 The Fundamental Spaces of a Matrix
7.4 The DimensionTheorem and Its Implications
7.5 The Rank Theorem and Its Implications
7.6 The Pivot Theorem and Its Implications
7.7 The Projection Theorem and Its Implications
7.8 Best Approximation and Least Squares
7.9 Orthonormal Bases and the Gram-Schmidt Process
7.10 OR-Decomposition; Householder Transformations
7.11 Coordinates with Respect to a Basis

8.1 Matrix Representations of Linear Transformations
8.2 Similarity and Diagonalizability
8.3 Orthogonal Diagonalizability; Functions of a Matrix
8.4 Quadratic Forms
8.6 Singular Value Decomposition

본문내용

• Properties of bases
Thm 1. If S=5%..., Uk3 is a basis for a subspace V for IRM.
every VEV is a linear combination of His in S in exactly one way
e.g. (R=
vert yes fer.., en? > standard basis for IM

(Proof) Let V E Spans=V
Suppose 1 = A. V,+.. +QKDk
= b.v.++bkVk
+ (arb,)y,+..=+(ak-bx) Vk = 1-10 = 0
By linear independence, Ar-br=0, Az-be =0,", Ax-bx=0
a, = bi,: Ak=bk

Thm 2. Let s be a finite subset of a subspace V of (on
(1) If SpanS = V but S is not a basis for V one can obtained a bass for V by removing some vectors from s. ine. Some subset of S is a basis for V
(2) If S is linearly independent but not a basis for v then one can obtain a basis for V by adding some vectors to S. ine. I a basis for V containing S.

(Proof) ff for (2) is almost some proof as the one for the exitence of a basis.
Since S is linearly dependent, ani which is a linear combination of in,: Vin
Say Dis = C.O. +-+ Cir Dir
Then. Claim: Span SliDi3 = V
For weV, say y=as. ++0.?k

참고 자료

Howard Anton, Robert C. Busby - Contemporary Linear Algebra-John Wiley & Sons (2002)