Linear Transformations (线性变换)

Transformation transforms an input to an output. In other words, a transformation is a function.

E.g., $f(x) = 2x$ is a transformation that transforms input $x$ to output $2x$.

A transformation $T$ that satisfies the following linearity property is called a linear transformation:

1. $T(x + y) = T(x) + T(y)$
2. $T(cx) = cT(x)$

Matrix as a Linear Transformation

General Discussion

The equition $A\vec{x} = \vec{b}$ can be viewed as a linear transformation that transforms input $\vec{x}$ to output $\vec{b}$, and the matrix $A$ represents this linear transformation, in other words, $A$ is a function.

A transformation $T$ from $\R^n$ to $\R^m$ is a function that takes an $n$-dimensional vector as input and produces an $m$-dimensional vector as output. $R^n$ is called the domain (定义域) of $T$, and $R^m$ is called the codomain (值域) of $T$.

For a given matrix $A$, we can define a linear transformation $T$ from $\R^n$ to $\R^m$ as $T(\vec{x}) = A\vec{x}$.

Copyright

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