Contents

3.1 Matrix Transformations

Functions

The Margalit text suggests thinking first of a function like

\[f(x)=x^2\]

except that we are now going to use a vector \(\vec x\) as the input. Our output will be a vector \(\vec y\), too. A transformation looks very similar.

\[T\left(\vec x\right) = \vec y\]

Example

Consider the transformation \(T\left(\vec x\right)\) accomplished by mutliplication of the vector \(\vec x\) by the matrix

\[\begin{split}A = \left[\begin{array}{rrrr}-2&1&3&1\\-2&1&2&0\\\end{array}\right]\end{split}\]
A = [-2 1 3 1 ; -2 1 2 0 ]

A =

    -2     1     3     1
    -2     1     2     0

Domain

The domain of \(T\left(\vec x\right)\) consists of all possible vectors that can be multiplied by \(A\), in this case, all possible 4-component vectors. The domain for \(T\left(\vec x\right)\) is \(\mathbb R^4\).

x1 = [1 ; 0 ; -3 ; 2];
x2 = [4 ; 3 ; -2 ; -1];

A * x1
A * x2

ans =

    -9
    -8


ans =

   -12
    -9

Codomain

When we multiply \(A\vec x\), the output is a 2-component vector, so the codomain is \(\mathbb R^2\). The range is subset of the codomain. The codomain is the vector space where the vectors live. The range is a subset of that vector space. Sometime the range subset is the entire codomain. If so, the transforamtion is called onto.

Consider an arbitrary vector in \(\mathbb R^2\).

\[\begin{split}\vec y = \left[\begin{array}{r}-2\\1\\\end{array}\right]\end{split}\]

Let’s create the augmented matrix and find the preimage.

y = [-2 ; 1]
rref([A,y])

y =

    -2
     1


ans =

    1.0000   -0.5000         0    1.0000   -3.5000
         0         0    1.0000    1.0000   -3.0000

Because the linear system is consistent, we know that \(\vec y\) is in the range of \(T\left(\vec x\right)\). We will dig deeper into these concepts in the next section.

Chapter 3 3.2 One-to-one and Onto Transformations