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Section 7.1 Transformations from \(\R^n\) to \(\R^n\)

Remember how functions \(f\colon X\to Y\) work. For each \(x\in X\text{,}\) the value of \(f(x)\in Y\) is defined. Typically this is done with a formula. For example, if \(f(x)=x^2\text{,}\) then for \(x=3\) we compute \(f(x)=f(3)=3^2=9\text{.}\) What is the set \(X\) in this case? So far, we don't know. It might be the nonnegative integers \(\N\) or the real numbers \(\R\text{,}\) so when defining a function it is important to understand \(X\) and \(Y\) as part of the definition.

Now consider defining \(g\colon X\to Y\) as \(g(x)=\sqrt x \text{.}\) If we try to use \(X=\R\text{,}\) there is a problem, since negative real numbers don't have (real) square roots, and we must be able to evaluate \(g(x)\) for all \(x\in X\text{.}\) If we use \(X=\N\) we don't have this problem, but we note that \(g(2)=\sqrt2\) is not in \(\N\text{.}\) If we use \(Y=\R\text{,}\) then there are no problems, so we can use \(g\colon\N\to\R\text{.}\) Every function has sets \(X\) and \(Y\) as part of its definition, and it is important to be aware of them, even if only implicitly. The set \(X\) is the domain and the set \(Y\) is the codomain.

Subsection 7.1.1 Matrix transformations

Transformations are functions where the domain is \(\R^n\) and codomain is \(\R^m\text{.}\)

Example 7.1.1. Transformations between \(\R^2\) and \(\R^3\).
  • \(T_1\colon\R^3\to\R^2\) defined by \(T((x,y,z))=(x+y,y+z).\)
  • \(T_2\colon\R^2\to\R^3\) defined by \(T((x,y))=(x+y,x-y,xy).\)

Using Example 7.1.1, evaluate \(T_1((1,2,3))\) and \(T_2((1,2))\text{.}\)

Solution
\begin{gather*} T_1((1,2,3))=(1+2,2+3)=(3,5)\\ T_2((1,2))=(1+2,1-2,1\cdot2)=(3,-1,2) \end{gather*}
Definition 7.1.3. Matrix transformation.

Let \(A\) be an \(m\) by \(n\) matrix, and let \(\vec x\) be a vector in \(\R^n\text{.}\) Then the matrix transformation \(T_A\colon \R^n\to \R^m\) is defined by

\begin{equation*} T_A(\vec x)=A\vec x\text{.} \end{equation*}
Example 7.1.4. A matrix transformation from \(\R^3\) to \(\R^2\).

Let

\begin{equation*} A=\begin{bmatrix}1\amp1\amp0\\ 0\amp1\amp1\end{bmatrix}\text{.} \end{equation*}

Then, using \(\vec x=(x,y,z)\text{,}\)

\begin{equation*} T_A(\vec x) = T_A((x,y,z)) = A \begin{bmatrix}x\\y\\z\end{bmatrix} =\begin{bmatrix}1\amp1\amp0\\ 0\amp1\amp1\end{bmatrix} \begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}x+y\\y+z\end{bmatrix}\text{,} \end{equation*}

and so \(T_A((x,y,z))=(x+y,y+z)\text{.}\)

As can be seen from this example, when writing transformations in the form \(T(\mathbf x)\) it is customary to write vectors horizontally, while when using matrix multiplication it is customary to write them vertically. This slight abuse of notation normally does not cause confusion. Compare this example with Example 7.1.1.

The two properties may be written as two easily verifiable equations:

  • \(\displaystyle A(\vec x+\vec y)=A\vec x+A\vec y\)
  • \(\displaystyle A(r\vec x)=rA\vec x\)

Transformations that satisfy the two properties in Theorem 7.1.5 are especially important.

Definition 7.1.6. Linear transformations.

A transformation \(T\colon \R^n\to\R^m\) is a linear transformation if it satisfies

  • \(\displaystyle T(\vec x+\vec y)=T(\vec x)+T(\vec y)\)
  • \(\displaystyle T(r\vec x)=rT(\vec x)\)

for any vectors \(\vec x\) and \(\vec y\) in \(\R^n\) and any scalar \(r\text{.}\)