본문 바로가기

Computer Science/Linear Algebra

08 Functions and linear transformations

728x90

A more formal understanding of functions


image
  • Range: subset of the codomain that the function actually maps to
image

Vector transformation


image

Linear transformations


image

Let's see this Transformation is a lienar treanformation.

image

We have to check the conditions.

image

This conficts with the requirement for a linear tranformation. So this is not a linear transformation!

Matrix vector products as linear transformations


image

By multiplying vector x times a, It can create a mapping from $R^n$ to $R^m$.

image

Linear transformations as Matrix vector products


Matrix multiplication or matrix products with vectors is always a linear tranformation!!

image

)

image

This satisfies all the definition of lienar transformation!

Any linear transformation can be represented by a matrix product!!

Image of a subset under a transformation

im(T): Image of a transformation


Let's say we have some set V in $R^n$.

Then we can say that like this.

image

)

image

)

image

Image of $R^n$ and I: T($R^n$) image of T in (T) is same with the column space of A.

image

Preimage of a set

Preimage and kernel example


image

)

image

Kernel of T : Ker(T) = {x $\in \mathbb R^2 \space$ | T($\vec {x}$) = {$\vec {0}$}

Sums and scalar multiples of linear tranformations


image
728x90

'Computer Science > Linear Algebra' 카테고리의 다른 글