08 Functions and linear transformations

A more formal understanding of functions

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

Vector transformation

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Linear transformations

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Let's see this Transformation is a lienar treanformation.

We have to check the conditions.

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

Matrix vector products as linear transformations

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

Linear transformations as Matrix vector products

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

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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

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

Then we can say that like this.

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

Preimage of a set

Preimage and kernel example

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

Sums and scalar multiples of linear tranformations

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