Linear algebra notes

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MATRIX

  • the essence of matrix - linear transformation
    • 2-by-2 matrix - 2-D space
    • 3-by-3 matrix - 3-D space
    • not square matrix
  • meaning of matrix multiplication Ax = b
    • linearly transform x in the manner called A resulting in B
    • A itself can be interpreted as a transformation of identity matrix, or unit basis factor in the space.
  • INVERSE MATRIX undo the linear transformation conducted by the original matrix
  • TRANSPOSE MATRIX
  • DETERMINANT det(A)
    • measure the change of linear transformation through show the change in area/volume of parallelogram/parallelepiped formed by the basis.
  • DOT product of vector & matrix production ==> projection
    • use symmetry to explain matrix & projection
  • CROSS production of vector and determinant ==> area/volume
    • for 2×2 matrix, det(A) = area of parallelogram = $\vec{a}\times\vec{b}$
    • for 3×3 matrix, det(A) = Volume of parallelepiped = $|\vec{a}\cdot ( \vec{b}\times\vec{c})|$, here $( \vec{b}\times\vec{c})$ shows a vector perpendicular to the plane spaned by b and c, whose length equals to the area. a projects on $\vec{b}\times\vec{c}$ and times the length of $\vec{b}\times\vec{c}$

VECTOR SPACE

  • VECTOR SPACE in linear algebra is actually a larger and more abstract idea rather than a simple arrow in space.
  • not necessarily in 3 dimensions.
  • any numbers pile together and form a . From the aspect of computer science, vector just means a set of numbers. ==> subsets fulfilling the same condition - subspace
  • the ultimate characteristic of vector space - it is linear
  • the characteristic of ‘linear’ - closed under addition and scalar multiplication(==> zero vector is included).
  • RANK is the dimension of the vector space of the linear transformation by a matrix
  • NULL SPACE where space high dimension is transformed to zero (the original point)
  • span of thos transformed basis vectors gives you all the possible outputs, which is called column space.

Reference - MATH2111 Dr. Ho’s notes

找到了MATH2111 Dr. Ho的note，虽然概念都是一样的，整理一下他的讲课思路看看对自己的理解有没有帮助

L1

Linear equations -> Linear system -> Augmented Matrix -> Elementary Row Operations × 3

(to solve equations) REF & RREF -> pivot position & pivot column -> (steps for RREF) -> Free variable -> （Particular solution & (General solution))

L2

Column vector in $R^n$ -> Linear conbination -> spanning

Matrix multiplication (这里也出现得莫名其妙似乎没有前因后果，但是好歹后面还有个 $A\vec{v}=\sum_{i=0}^n a_iv_i$ 算是应用了一下)

L3

(Equivalent Forms)

Solution structure of a linear system - Homogeneous system is $A\vec{x}=\vec{0}$ ; Trivial solution is $\vec{x}=\vec{0}$ ; Non-trivial solution is $\vec{x}\neq\vec{0}$ ;

Non-homogeneous system is $A\vec{x}=\vec{b} $ where $\vec{b}\neq\vec{0}$ ; Genral forms of the solution: $\vec{p}+\vec{v_h}$ where $\vec{p}$
is a particular solution and $\vec{v_h}$ is the genral form of solutions of $A\vec{x}=\vec{0}$

Linear Independence

L4

Linear transformation - a mapping T from a vector space to another vector space

Standard unit vectors, standard basis, standard matrix for linear transformation T

Composition

Transpose

L5

One-to-One onto

invertible, inverse matrix, identity transformation, 2×2, 3×3 determinant -> Definition of determinant

cofactor of A (i,j) $ C{ij}=(-1)^{i+j}det A{ij} $

L6

Classical Adjoint Matrix, elementary matrix, cramer’s rule(not mentioned in 2350)

L7

Null space $Nul A = {\vec{x}:\vec{x} \space is\space in\space R^n and A\vec{x}=vec{0}}$

a. we may regard null space of A as a family which contains at least one member

L8

basis

standard basis for Rn and Pn

L9

Row space

Coordinate system

Change of Coordinate Matrix

L10

Dimension

Eigenvector Eigenvalue Eigenspace

L11

similar matrix
diagonal matrix, diagonalizable matrix

General Solution to differential equation

1st order separable differential equation

L12

Inner product - Dot product

Length, distance

Orthogonal

L13

Orthogonal Projection

Remarks

写于191004

这两星期学linear algebra 学得有点懵逼，看了一下3b1b的视频，总结一下现在学到的知识。

写于191112

这门MATH的前半部分linear algebra终于告一段落。midterm的成绩自己也比较满意（毕竟题目简单）。

进入到后半部分differential equation，终于领会到这两个内容放在同一门课学的意义，也很感谢prof精彩的串讲（还有不时出现的Fibonacci），感受到了这两个知识模块内在的数学联系。同时对另外一堂multivariable calculus也有很大帮助。

当然难一点的题我还是不会，不过作为engineer最重要的是会用嘛（自我麻痹）。