#400#ML_Engineer_Basic#410#Mathematics#410.00#Linear_Algebra#410.00 a#Vectors_Properties

#Vector #Vector_Operation

Linear Combination

Operations that combine the product and addition of scalars to get a new vector

Linear Independence

Given vectors v1,v2,...,vnv1,v2,...,vn, they are linearly independent if the following equation holds true only when c1=c2=...=cn=0c1=c2=...=cn0:

c1v1+c2v2+...+cnvn=0c1v1+c2v+...+cvn=0

Here, c1,c2,...,cncc2,...,cn are scalar coefficients, and 00 represents the zero vector.

Linearly Dependece

Given vectors v1,v2,...,vnv1,v...,v, they are linearly dependent if there exists a set of scalar coefficients c1,c2,...,cnc1,c2,...,cn, not all zero, such that:

c1v1+c2v2+...+cnvn=0c1v1+c2v2+...+cnvn0

Linear dependence means at least one vector in the set can be expressed as a combination of the others.

Representation of vector space

Subspace

1. Closure: The sum and scalar multiples of any vectors in the subspace must also belong to the subspace.

2. Contains Zero Vector: The subspace always includes the zero vector.

3. Operations Follow Vector Space Rules: All vectors in the subspace must adhere to the rules of vector addition and scalar multiplication.

Vector Basis

It's a set of vectors in a vector space that is both linearly independent and spans the entire space.

Standard Basis Vector

For an nn-dimensional space, there are nn standard basis vectors.

Example in 3D: e1=(1,0,0)e1=(1,0,0), e2=(0,1,0)e2=(0,1,0), e3=(0,0,1)e3=(0,0,1).

Vector Norm (Magnitude or Length)

p-norm

Let $p>=1$ be a real number. The$p-norm$ (also called$l^p-norm$) of vector $x=(x_1,\dots ,x_n)$is

Used primarily to obtain the shortest distance in a two-dimensional plane