Functions and Linear Transformations

Functions

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

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

$$i) f:x \rightarrow x^2; \mathbb{R} \rightarrow \mathbb{R} \newline ii) f:(x_1,x_2) \rightarrow (x_1+x_2, x_2-x_1,x_1*x_2); \mathbb{R}^2 \rightarrow \mathbb{R}^3 (Higher\hspace{.1cm}dimension)$$

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:

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

Link:

• https://www.khanacademy.org/math/linear-algebra/matrix-transformations/linear-transformations/v/a-more-formal-understanding-of-functions
• https://www.khanacademy.org/math/linear-algebra/matrix-transformations/linear-transformations/v/linear-transformations
• 3Blue1Brown: Linear transformations and matrices