Functions and Linear Transformations


Functions are mathematical entities that maps/associates input value to outputs.

In the (ii) example: R2R^2: domain R3R^3: co-domain Range: subset of co-domain which consists of actual points/value where the function maps to.

i)f:xx2;RRii)f:(x1,x2)(x1+x2,x2x1,x1x2);R2R3(Higherdimension)i) f:x \rightarrow x^2; \mathbb{R} \rightarrow \mathbb{R} \newline ii) f:(x_1,x_2) \rightarrow (x_1+x_2, x_2-x_1,x_1*x_2); \mathbb{R}^2 \rightarrow \mathbb{R}^3 (Higher\hspace{.1cm}dimension)

Linear Transformations

Transformation is another term for function which moves a vector in space from A to B. Suppose a 10x10 grid needs to be transformed linearly. Then it must follow two rules:

  • The transformed grid should still be consisted of only lines (i.e No curves)

  • Origin remains fixed

Technically, a transformation is called linear if it follows these rules:

T:RnRmwherev,wϵRni)T(v+w)=T(v)+T(w)ii)T(cv)=cT(v)T: \mathbb{R}^n \rightarrow\mathbb{R}^m where \hspace{.1cm} \vec{v}, \vec{w} \epsilon\mathbb{R}^n \newline i)\hspace{.1cm} T(\vec{v} + \vec{w}) = T(\vec{v}) + T(\vec{w}) \newline ii) \hspace{.1cm} T(c\vec{v}) = cT(\vec{v})


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