IMPERIAL COLLEGE OF LONDIN
IMPERIAL COLLEGE OF LONDIN
The first 4 books (by Strang, Lang, etc) are the Masterpieces.
Strang’s MIT OpenCourse:
The Big Picture of Linear Algebra 线性代数 (MIT Open-Courseware by the famous Prof Gilbert Strang)
A.X = B
A(m, n) is a matrix of m rows, n columns
4 sub-Vector Spaces:
Column Space , dim = r = rank (A)
Row Space , dim = r = rank (A)
Nullspace , dim = n – r
Left Nullspace , dim = m – r
Abstract Vector Spaces 向量空间
Any object satisfying these 8 axioms belong to the algebraic structure of Vector Space: eg. Vectors, Polynomials, Functions, …
Note: “Vector Space” + “Linear Map” = “Category”
Eigenvalues & Eigenvectors (valeurs propres et vecteurs propres) 特征值/特征向量
[ Note: “Eigen-” is German for Characteristic 特征.]
Important Trick: (see Monkeys & Coconuts Problem)
If a transformation A is linear, and the “before” state and “after” state of the “vector” v remain the same (keep the status-quo) , then : Eigenvalue
Try to compute:
is more difficult than this diagonalized equivalent matrix:
Note: This is the secret of the Google computation of diagonalized matrix of billion columns & billion rows, where all the bjk are the “PageRanks” (web links coming into a particular webpage and links going out from that webpage).
The Essence of Determinant (*): (行列式)
(*) Determinant was invented by the ancient Chinese Algebraists 李冶 / 朱世杰 /秦九韶 in 13th century (金 / 南宋 / 元) in《天元术》.The Japanese “和算” mathematician 关孝和 spread it further to Europe before the German mathematician Leibniz named it the “Determinant” in 18th century. The world, however, had to wait till the 19th century to discover the theory of Matrix 矩阵 by JJ Sylvester (Statistical Math private Tutor of Florence Nightingale, the world’s first nurse) closely linked to the application of Determinant.
[NOTE] 金庸 武侠小说 《神雕侠女》里 元朝初年的 黄蓉 破解 大理国王妃 瑛姑 苦思不解的 “行列式”, 大概是求 eigenvalues & eigenvectors ? 🙂
Linear transformations are intuitively those maps of everyday space which preserve “linear” things. Specifically, they send lines to lines, planes to planes, etc., and they preserve the origin.
(One which does not preserve the origin is very similar but has a different name; see Affine Transformation)
Let A matrix, vector x, λ eigenvalue
1. Right Eigenvectors and eigenvalues:
A.x = λx
2. Left Eigenvectors and eigenvalues:
x.A = λx
However, be careful that:
If we want to find the left eigenvector associated with the eigenvalue 5, then we find the eigenvector .
This would lead us to see that:
(-1 1 -1).A = (-5 5 -5) = 5. (-1 1 -1)
So, in this example, the eigenvalue 5 has different left and right eigenvectors:
(-1 1 -1) & (1 1 1) respectively.
Remark 1: However, the nice fact about matrices is that always :
left eigenvalue = right eigenvalue.
So we just simply call eigenvalue for short.
《Math Bytes》by Tim Charter
Princeton University Press
[NLB #510 CHA]