Linear Algebra - 4

Section 1.8 Linear Transformations

A transformation : $T : \R^n\to \R^m$

domain : $\R^n$ . codomain : $\R^m$ . $T(x)$ is the image of $x$ . The set of $T(x)$ is called range.

Matrix Transformation : $x\mapsto Ax$

shear transformation:

$\begin{bmatrix} 1&2\\ 0& 1 \end{bmatrix}$

Linear Transformation

A transformation is linear iff:

$T(\vec u+\vec v)=T(\vec u)+T(\vec v)$
$cT(\vec u)=cT(\vec u)$

which can be simplified to:

$T(c\vec u+d\vec v)=cT(\vec u)+dT(\vec v)$

Matrix transformation is a kind of linear transformation.

If $T$ is a linear transformation, then $T(0)=0$

Section 1.9 The Matrix of Linear Transformation

Let $T:\R^n\to \R^m$ , there exists a unique matrix $A$ such that $T(x)=Ax$

$A=[T(e_1)\dots T(e_n)]$

where $e_j$ is the $j$ th column of the identity matrix in $\R^n$

A mapping $T:\R^n\to \R^m$ $T : R^{n} \to R^{m}$ is said to be onto $\R^m$ $R^{m}$ if each $b$ $b$ in $\R^m$ $R^{m}$ is the image of at least one $x$ $x$ in $\R^n$ $R^{n}$
- The columns of $A$ span $\R^m$
A mapping $T:\R^n\to\R^m$ $T : R^{n} \to R^{m}$ is said to be one-to-one if each $b$ $b$ in $\R^m$ $R^{m}$ is the image of at most one $x$ $x$ in $\R^n$ $R^{n}$
- All the columns of $A$ are linearly independent.

Existence and Uniqueness Questions:

Is it a map from $\R^4$ to $\R^3$ : Check if each row has a pivot position. (If $Ax=b$ is consistent for every $b$ ) This is called onto.

Is it a one-to-one mapping? (Check if $x$ is unique) Check if there is a free variable, that there are multiple solutions $x$ .

One-to-One have no relevant to onto! Onto see the codomain, while one-to-one see the domain !!!

Which is equated to $T(x)=0$ has only trivial solution(iff)

Review : Existence of Solutions

For each $b$ in $\R^m$ , the equation $Ax=b$ has a solution
Each $b$ in $\R^m$ is a linear combination of the columns of $A$
The columns of $A$ span $\R^m$
$A$ has a pivot position in every row

The columns of matrix $A$ are linearly independent iff the equation $Ax=0$ has only the trivial solution.

a pivot in every column guarantees at most one solution; a pivot in every row guarantees at least one