这是自学由 MIT 的 W.Gilbert Strang 教授开设的 18.06 Linear Algebra 课程时所作的笔记摘要
¶Lecture 1
$n$ Linear equations, $n$ unknowns
Row Picture
Column Picture
matrix form
Row Picture
$2x - y = 0$
$-x + 2y = 0$
Matrix $\boldsymbol A$
Column Picture
linear combination of columns
$ x\begin{bmatrix} 2 \\ -1 \end{bmatrix} + y\begin{bmatrix} -1 \\ 2 \end{bmatrix} =\begin{bmatrix} 0 \\ 3 \end{bmatrix}\\ $ $ 1\begin{bmatrix} 2 \\ -1 \end{bmatrix} + 2\begin{bmatrix} -1 \\ 2 \end{bmatrix} =\begin{bmatrix} 0 \\ 3 \end{bmatrix}\\ $$\boldsymbol A$ times $\boldsymbol x$ is a combination of the columns of $\boldsymbol A$
$ x\begin{bmatrix} 2 \\ -1 \\ 0 \end{bmatrix} + y\begin{bmatrix} -1 \\ 2 \\ -3 \end{bmatrix} + z\begin{bmatrix} 0 \\ -1 \\ 4 \end{bmatrix} =\begin{bmatrix} 0 \\ -1 \\ 4 \end{bmatrix}\\ $$x = 0, y = 0, z = 1$
Can I solve $\boldsymbol A \boldsymbol x = \boldsymbol b$ for every right-hand side $\boldsymbol b$?
Do the linear combinations of the columns fill three dimensional space?
singular case: the matrix would be invertible
non-singular case: invertible and beautiful
¶Lecture 2
Elimination Success or Failure
Back-Substitution
Elimination matrices
Matrix multiplication
¶Lecture 3
Matrix multiplication
Inverse of $\boldsymbol A$ $\boldsymbol A \boldsymbol B$ $\boldsymbol A^\mathrm{T}$
Gauss-Jordan / find $\boldsymbol A^{-1}$
¶Lecture 4
Inverse of $\boldsymbol A \boldsymbol B$, $\boldsymbol A^\mathrm{T}$ (What’s the inverse of a product?)
Product of elimination matrices
$\boldsymbol A = \boldsymbol L \boldsymbol U$ (no row exchanges)
¶Lecture 5
$\boldsymbol P \boldsymbol A = \boldsymbol L \boldsymbol U$
Permutation / Transposes
Vector spaces
and subspaces
¶Lecture 6
Vector spaces and Subspaces
Column Space of $\boldsymbol A$
Nullspace of $\boldsymbol A$
: Solving $\boldsymbol A \boldsymbol x = \boldsymbol b$
¶Lecture 7
Computing the nullspace ( $\boldsymbol A \boldsymbol x = \boldsymbol 0$ )
Pivot variables - free variables
Special Solutions - $\text{rref }(\boldsymbol A) = \boldsymbol R$
¶Lecture 8
Complete solution of $\boldsymbol A \boldsymbol x = \boldsymbol b$
Rank $r \ \ \ \ x = x_p + x_n$
$r = m : $ solution exists
$r = n : $ solution is unique
¶Lecture 9
Linear independence
Spanning a space
BASIS and dimension
¶Lecture 10
Four Fundamental Subspaces
(for matrix $\boldsymbol A$)
column space : $\text{C}(\boldsymbol A)$ in $\mathbb{R}^{m}$
nullspace : $\text{N}(\boldsymbol A)$ in $\mathbb{R}^{n}$
row space : = all combinations of rows = all combs of columns of $\boldsymbol A^\mathrm{T} = \text{C}(\boldsymbol A^\mathrm{T})$ in $\mathbb{R}^{n}$
nullspace of $\boldsymbol A^\mathrm{T}$ : = $\text{N}(\boldsymbol A^\mathrm{T})$ in $\mathbb{R}^{m}$
¶Lecture 11
Bases of new vector spaces
Rank one matrices
Small world graphs
¶Lecture 12
Graphs & Networks
Incidence Matrices
Kirchhoff’s Laws
¶Lecture 13
Review for Exam 1
Emphasizes Chapter 3
¶Lecture 14
Orthogonal vector & Subspaces
nullspace $\perp$ row space
$\text{N}(\boldsymbol A^\mathrm{T} \boldsymbol A) = \text{N}(\boldsymbol A)$
¶Lecture 15
Projections!
Least squares
PROJECTION MATRIX
¶Lecture 16
Projections
Least squares and
best straight line
¶Lecture 17
Orthogonal basis $q_1, q_2, \cdots, q_n$
Orthogonal matrix $\boldsymbol Q$ : square
Gram-Schmidt $\boldsymbol A \to \boldsymbol Q$
Orthonormal vectors $q^\mathrm{T}_i q_j = \begin{cases} 0 & \text{if } i \neq j, \\ 1 & \text{if } i = j.\end{cases}$
¶Lecture 18
Determinants $\text{det } \boldsymbol A$
Properties 1,2,3,4-10
$\pm$ signs
¶Lecture 19
Formula for $\text{det } \boldsymbol A$
Cofactor furmula
Tridiagonal matrices
¶Lecture 20
Formula for $\boldsymbol A^{-1}$
Cramers Rule for $\boldsymbol x = \boldsymbol A^{-1} \boldsymbol b$
$| \text{Det } \boldsymbol A|$ = Volume of box
