Day #21 - Data Science - Maths Basics - Vectors and Matrices

Matrix - Combination of rows and columns
Check for Linear Dependence - R2 = R2 - 2R1, When one of the rows is all zeros it is linearly dependent
Span - Linear combination of vectors
Rank - Linearly Independent set

Good Related Read - Span

Vector Space - Space of vectors, collection of many vectors
If V,W belong to space, V+W also belongs to space, multiplied vector will lie in R Square
If the determinant is non-zero, then the vectors are linearly independent. Otherwise, they are linearly dependent

Vector space properties

• Commutative  x+y = y+x
• Associative (x+y)+z = x+(y+z)
• Origin vector - Vector will all zeros, 0+x = x+0 = x
• Additive (Inverse) - For every X there exists -x such that x+(-x) = 0
• Distributivity of scalar sum, r(x+s) = rx+rs
• Distributivity of vector sum, r(x+s) = rx+rs
• Identity multiplication, 1*x = x

Subspace
Vector Space V, Subset W. W is called subspace of V
Properties
W is subspace in following conditions

• Zero vector belongs to W 
• if u and v are vectors, u+v is in W (closure under +)
• if v is any vector in W, and c is any real number, c.v is in W

Subset S belongs to V can be represnted as linear combination
 v = r1v1+ r2v2+... rkvk
v1,v2 distinct vectors from S, r belongs to R

Basis - Linearly Independent spanning set. Vector space is called basis if every vector in the vector space is a linear combination of set. All basis for vector V same cardinality

Null Space, Row Space, Column Space
Let A be m x n matrix

• Null Space - All solutions for Ax = 0, Null space of A, denoted by Null A, is set of all homogenous solution for Ax=0
• Row Space - Subspace of R power N spanned by row vectors is called Row Space
• Column Space -  Subspace of R power N spanned by column vector is called Column Space

Norms - Measure of length and magnitude

• For (1,-1,2), L1 Norm = Absolute value = 1+1+2 = 4
• L1 - Same Angle
• L2 - Plane
• L3 - Sum of vectors in 3D space
• L2 norm (5,2) = 5*5+2*2 = 29
• L infinity - Max of (5,2) = 5

Orthogonal - Dot product equals Zero
Orthogonality - Linearly Independent, perpendicular will be linearly independent
Orthogonal matrix will always have determinant +/-1

Map of Mathematics.

Enlarge the figure to see all the wonderful areas for exploration and imagination. Which topic might you find most fascinating?

By Dominic Walliman, @DominicWalliman, Source: https://t.co/mNu0hWzFGW, Used with permission. pic.twitter.com/kx1azWIhle

— Cliff Pickover (@pickover) August 22, 2022

Differential Equations - Notes - Link

Lectures - Link

Course Notes - Link

Happy Learning!!!