Try each of the following matrices to determine what each transformation accomplishes. (Type pi into GeoGebra to get .)
Match the description of each transformation with the matrix that induces it.
Matrix Transformations
Functions from into
In the past you have worked with functions . Most of the time such functions were defined algebraically. For example, we can define by . This function takes a number in the domain () and maps it to the square of the number in the codomain (also ). Previously, you might have visualized function by looking at its graph, the set of all points of the form in . In this course, we will find it more useful to look at functions diagrammatically. For instance, the diagram below shows that maps 2 to 4. We say that 4 is the image of 2 under .
We will now consider functions that map into . We will refer to such functions as transformations. There are two ways of thinking of transformations. A transformation can take a vector in and map it to a vector in , or it can map a point in to a point in . We will think of transformations as acting on vectors or points interchangeably because every point of can be interpreted as the tip of a vector in . Matrix multiplication will provide us with initial tools for defining some transformations.
Examples of Matrix Transformations
Consider the matrix . The product of with a vector is a vector. We can define a transformation by . This transformation can be applied to every vector of . We will look at what it does to five vectors.
Even after looking at a handful of vectors it is often difficult to tell what the transformation actually accomplishes. This is why sometimes looking at points instead of vectors can be beneficial. If we consider every point in the left grid below as a tip of a vector, we can apply the transformation to each point to obtain the grid on the right.
Applying to a grid of points helps us see that the entire plane was sheared by the transformation.
We can also analyze the action of algebraically. Start by finding the image of a generic vector . We immediately see that the component of the vector remains unchanged. We also see that the component increases (or decreases) by an increment that depends on . When considering as a transformation acting on points, we see that points located 1 unit above the -axis, get shifted to the right by 0.5. Points located 2 units above, get shifted to the right by 1. The higher the point, the greater the shift. Points with negative -coordinates get shifted to the left. In this fashion shears the entire plane.
Now that we have seen the effect of functions defined via matrix multiplication, we can better appreciate the term transformation, as such functions distort the domain and the shapes located in it. The following Exploration will help you visualize this.
Make your own shape by moving points in the left pane. (You can also move the
entire figure by clicking and dragging the whole polygon.) The images of the points
and the polygon under the transformation induced by are shown on the
right.
A matrix induces a transformation from into . An matrix can be multiplied by an vector on the right, with the resulting product being an vector. Therefore we can use an matrix to define a transformation by .
Linearity of Matrix Transformations
Restating properties item:matrixproperties1 and item:matrixproperties4 of Section Matrix Multiplication in terms of matrix-vector multiplication gives us
\begin{eqnarray} \label{eq:linearityConstant} k(A\vec{v})&=&A(k\vec{v})\\ A(\vec{v}+\vec{w})&=&A\vec{v}+A\vec{w}\label{eq:linearityAdd} \end{eqnarray}
These two properties of matrix multiplications translate into analogous properties of matrix transformations. Suppose is a matrix transformation, then for all vectors , in and all constants in , \begin{equation} \label{eq:matrixTransProp1} T(k\vec{v})=kT(\vec{v}) \end{equation} \begin{equation} \label{eq:matrixTransProp2} T(\vec{v}+\vec{w})=T(\vec{v})+T(\vec{w}) \end{equation}
In general, any transformation that satisfies (eq:matrixTransProp1) and (eq:matrixTransProp2) is called a linear transformation. As we have just seen, all matrix transformations are linear. We will study linear transformations in depth later in this chapter.
Where did Go?
In this section we will look at the images of standard unit vectors under a matrix transformation, and discuss why this information is helpful.
Let . Find the following products:
Let be a matrix transformation induced by , then we can say that maps , and to the first, second and third columns of , respectively.
This nice property is not limited to transformations induced by square matrices. Let be a linear transformation induced by We will examine the effect of on the standard unit vectors , and .
Observe that the image of is the first column of , the image of is the second column of , and the image of is the third column.
We formalize our findings in Exploration exp:imagesOfijk as follows.
In general, the linear transformation , induced by an matrix maps the standard
unit vectors to the columns of . We summarize this observation by expressing
columns of as images of vectors under .
\begin{equation*} \label{eq:matlintransIntro} A=\begin{bmatrix} a_{11} & a_{12}&\dots &a_{1n}\\ a_{21}&a_{22} &\dots &a_{2n}\\ \vdots & \vdots &\ddots &\vdots \\ a_{m1}&\dots &\dots &a_{mn} \end{bmatrix} = \begin{bmatrix} | & |& &|\\ T(\vec{e}_1) & T(\vec{e}_2)&\dots &T(\vec{e}_n)\\ |&| & &| \end{bmatrix} \end{equation*}
Why is it that knowing the images of standard unit vectors under a matrix transformation is helpful? Consider the following example.
Example ex:imageOfBasisVectors illustrates that a matrix transformation is completely determined by where it maps the standard unit vectors. This is true because we can express every vector in as a linear combination of the standard unit vectors, then use (eq:matrixTransProp1) and (eq:matrixTransProp2) to find the image of .
Practice Problems
Let be a matrix transformation induced by the matrix . The GeoGebra window on
the left shows the domain of , with standard unit vectors and , and a vector . The
window on the right shows the codomain of , with the images of , and plotted. To
use this interactive, you can
- Change the entries of matrix ;
- Change vector by dragging its tip.
Choose your matrix . Visually verify the following claims:
- The image of is the first column of matrix .
- The image of is the second column of matrix .
Let . Complete the following statement by filling the blanks. Choose a different matrix , but keep vector the same. Does the above relationship still hold?
Change vector by dragging its tip. Observe the image of and its relationship to the images of and . Complete the following statement for all vectors in .
Show that a matrix transformation maps a line in to a line (or the origin) in .
2024-09-11 17:55:35 A
line in can be expressed as . (See Formula form:vectorlinend.)