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 $T(\vec{x}) = A\vec{x}$.

One-to-one and Onto (单射和满射)

One-to-one (单射)

A linear transformation $T$ from $\R^n$ to $\R^m$ is called one-to-one if for every $\vec{b}$ in the codomain $\R^m$, there is at most one $\vec{x}$ in the domain $\R^n$ such that $T(\vec{x}) = \vec{b}$.

A linear transformation $T$ from $\R^n$ to $\R^m$ is one-to-one if and only if the columns of the corresponding matrix $A$ are linearly independent, which is equivalent to saying that $A$ has a pivot in every column .

Onto (满射)

A linear transformation $T$ from $\R^n$ to $\R^m$ is called onto if for every $\vec{b}$ in the codomain $\R^m$, there is at least one $\vec{x}$ in the domain $\R^n$ such that $T(\vec{x}) = \vec{b}$.

A linear transformation $T$ from $\R^n$ to $\R^m$ is onto if and only if the columns of the corresponding matrix $A$ span $\R^m$, which is equivalent to saying that $A$ has a pivot in every row.

Discussion

If a linear transformation $T$ from $\R^n$ to $\R^m$ is one-to-one, then $n \leq m$; if it is onto, then $n \geq m$. This is intuitive when thinking about the "number" of elements in the domain and codomain, while it can be proved by the above definitions and the properties of the corresponding matrix $A$.

Copyright

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