The matrix of a linear transformation

The matrix of a linear transformation


What is a linear transformation

From our lesson on the image and range of linear transformations we learnt that a linear transformation is a technique in which a vector gets "converted" into another by keeping a unique element from each of the original vector and assigning it into the resulting vector. This process basically maps one vector space into another. There may be different kinds of transformations in mathematics but the case of a linear transformation in linear algebra is that which preserves linear combinations from the original expression to the resulting one, in simple words, it preserves the way addition and scalar multiplication operate from the original vector to the transformed vector. A linear transformation example can also be called linear mapping since we are keeping the original elements from the original vector and just creating an image of it.

Recall the matrix equation Ax=b, normally, we say that the product of AA and xx gives bb. Now we are going to say that A is a linear transformation matrix that transforms a vector x into a vector bb (we now call bb an image of a linear transformation of xx).

Properties of linear transformation
Figure 1: Linear transformation

In a sense AxAx is a function where if we plug in a vector, then it spits out another vector. If we call this function T(x)T(x), then T(x)=AxT(x)=Ax, where TT is the transformation. Note that T(x)T(x) is an image of xx since T(x)=bT(x)=b;


The matrix of a linear transformation

So, let us define once more a linear transformation in mathematical form:

Equation 1: Linear transformation of vector x
Equation 1: Linear transformation of vector x

Where T(x)T(x) is read as "the transformation of xx", xx is the initial vector, and AA is the transformation matrix. We will focus on this AA this is what we call: the standard matrix of a linear transformation.

During this lesson, we will focus on methods to solve for the standard matrix A when you are given a linear transformation of a vector.

Let us start by describing the three main methods we use to solve such problems and then we will jump into examples of such exercises.

How to find the standard matrix of a linear transformation


There are three methods we can use to obtain the standard matrix of a linear transformation:

  • Calculating the standard matrix using the transformed basis:

For a two dimensional basis, when using this method, we are provided with the unit vectors e1e_{1} and e2e_{2} and with the transformed basis vectors T(e1)T(e_{1}) and T(e2)T(e_{2}) to start with. If we are talking about a three-dimensional basis then we are provided with the unit vectors e1,e2e_{1}, e_{2} and e3e_{3} and the transformed vectors T(e1)T(e_{1}), T(e2)T(e_{2}) and T(e3)T(e_{3}). Then, the steps to follow to obtain the standard matrix are:

  1. Use the definition of a linear transformation: T(x)=AxT(x)=Ax and define the vector xx.
  2. Expand vector xx into its components until is written in terms of the unit vectors and its variables x1,x2,...,xnx_{1}, x_{2},..., x_{n}.
  3. Use the expansion of vector xx and perform the linear transformation T(x)=AxT(x)=Ax.
  4. Expand the transformation T(x)T(x) until is in terms of the transformed vectors T(e1)T(e_{1}), T(e2)T(e_{2}) , ... , T(en)T(e_{n}) and the variables x1,x2,...,xnx_{1}, x_{2},..., x_{n}.
  5. Then rewrite this expression from the transformation as the matrix multiplication AxAx.
  6. You have found the matrix AA.

Notice that the standard matrix found is composed of the transformed vectors T(e1)T(e_{1}), T(e2)T(e_{2}) , ... , T(en)T(e_{n}). Technically, you could very well skip this whole process and just write down the matrix using the provided transformed vectors, BUT, you need to know how they were derived (plus your teacher may not accept this solution without an explanation). So make sure you understand the whole process.
Examples of this method are found in example exercises 1 and 2 in our next section.

  • Finding the standard matrix geometrically:

To find a standard matrix geometrically means to find it using a cartesian coordinates graph, actually draw the vectors involved, see the transformations they go through and obtain from the result a new matrix: the standard matrix. The steps to follow to find a standard matrix geometrically for a two-dimensional basis are:

  1. Draw the standard basis e1e_{1} and e2e_{2} in the x1x2planex_{1}x_{2}-plane.
  2. Obtain the transformed vectors T(e1)T(e_{1}) and T(e2)T(e_{2}).
  3. Draw the transformed vectors using the information provided in the problem.
  4. Combine the transformed vectors to produce the standard matrix.

Notice that are providing the steps for a two-dimensional basis only, since we can a two-dimensional coordinate plane only (actually, you can also draw a three-dimensional plane, but it gets a bit complicated. Further than three dimensions we cannot draw them, and so, we use other techniques to solve for transformations with such bases).
For a step by step explanation on how to use this method, look at example 3 in our last section for this lesson.

  • Computing the standard matrix algebraically:

The algebraic computation in order to find the standard matrix of a linear transformation can be done easily when the problem provides the complete expression for the linear transformation T(x)=AxT(x)=Ax. In other words, once you have the initial vector x and the final vector AxAx, you can use these expressions to solve for the matrix AA.
The steps to follow in order to solve a problem this way are:

  1. Expand the given expression of the linear transformation resulting vector into an addition of vectors for each of the different variables inside.
  2. Factor out the variables x1,x2,...,xnx_{1}, x_{2},..., x_{n} from each of the column vectors.
  3. Form a matrix multiplication by rearranging the resulting column vectors from our last step into a matrix, and then multiplying it for the vector x which is the column vector containing all of the variables x1,x2,...,xnx_{1}, x_{2},..., x_{n}.
  4. The matrix that was formed in our last step is the standard matrix of the linear transformation.

An example of this process can be seen in the exercise example 4 in our next section.
Notice that this is probably the simplest methods to use, and if you are given all of the necessary information and have to be time-effective while solving an exercise on this, we recommend you to use this method.

Linear transformation matrix examples


During the first two examples we will be finding the Standard Matrix using a transformed basis, then we will see two examples on how to find the standard matrix geometrically. We finish off with an example where we will compute the standard matrix algebraically.

Example 1

Assume that TT is a linear transformation. Find the matrix representation of linear transformation TT (the standard matrix) if:

The matrix of a linear transformation
Equation 2: Conditions for the matrix transformation

Where the unit vectors are defined as follows:

The matrix of a linear transformation
Equation 3: Unit vectors e1 and e2

To answer this question we could literally just take the vectors resulting from the linear transformation of the unit vectors e1e_{1} and e2e_{2}, put them together into a matrix and that would be our standard matrix, but how does that work? Let us derive it then!

We know that a linear transformation of a matrix AA has the form: T(x)=AxT(x)=Ax, therefore, we need a vector xx to perform the transformation. For this case our vector xx must have two entries, and so, vector xx is (notice this is just any vector xx in two dimensions):

The matrix of a linear transformation
Equation 4: Vector x

We can expand this vector x into an addition of vectors and factor out their components in order to have the unit vectors which form part of it visible, just look at how we have expanded it below:

The matrix of a linear transformation
Equation 5: Vector x expansion (part 1)

And now, it can clearly be seen how the vector x is composed by the variables x1x_{1} and x2x_{2}, and the set of unit vectors e1e_{1} and e2e_{2}. Thus we can rewrite this last equation as:
Equation 6: Vector x expansion (part 2)


So we take the linear transformation of vector xx which goes as:
Equation 7: Linear transformation of vector x
Equation 7: Linear transformation of vector x


Notice how the linear transformation affects the unit vectors only, this is because those are the ones establishing the direction of the vector and a linear transformation usually changes that (remember it can rescale a vector too, but the most common cases it preserves its magnitude, but rotates it with respect of a certain axes). So, since we know what T(e1e_{1}) and T(e2e_{2}) are we can rewrite the transformation as:

The matrix of a linear transformation
Equation 8: Rewriting the linear transformation of vector x

Where the standard matrix of a linear transformation as the one represented in equation 2 is:

The matrix of a linear transformation
Equation 9: Standard matrix A

Example 2

Assume that TT is a linear transformation. Find the standard matrix of linear transformation TT if:

The matrix of a linear transformation
Equation 10: Conditions for the matrix transformation

where the unit vectors are defined as:

The matrix of a linear transformation
Equation 11: Unit vectors e1, e2and e3

A linear transformation of a matrix AA has the form: T(x)=AxT(x)=Ax, thus, we start by expanding the vector xx into its components once more:

The matrix of a linear transformation
Equation 12: Vector x expansion (part 1)

We can observe how vector xx is composed by the variables x1,x2x_{1}, x_{2} and x3x_{3}, and the set of unit vectors e1,e2e_{1}, e_{2} and e3e_{3}. Thus we can rewrite this last equation as:
Equation 13: Vector x expansion (part 2)
Equation 13: Vector x expansion (part 2)


So we take the linear transformation of vector x which goes as:
Equation 14: Linear transformation of vector x
Equation 14: Linear transformation of vector x


Since we know what T(e1e_{1}), T(e2e_{2}) and T(e3e_{3}) are we can rewrite the transformation as:

The matrix of a linear transformation
Equation 15: Rewriting the linear transformation of vector x

And so, the standard matrix is:

The matrix of a linear transformation
Equation 16: Standard matrix A


Example 3

Assume that TT is a linear transformation. Find the matrix of the linear transformation TT if TT: R2R2R^{2} \to R^{2} is a vertical shear transformation that maps e1e_{1} to e1+2e2e_{1}+2e_{2} and leaves e2e_{2} unchanged.
Let us first define the standard basis for this problem, given that we are talking of a 2-dimensional vector space R2R^{2}, then this means our standard basis unit vectors are:

The matrix of a linear transformation
Equation 17: Standard basis unit vectors

For this example we will use the steps summarized on the method to find a standard matrix geometrically:

1. \quad Draw the standard basis e1e_{1} and e2e_{2} in the x1x2planex_{1}x_{2}-plane.


The matrix of a linear transformation
Figure 2: standard basis e1 and e2 in the x1x2-plane.


2. \quad Obtain the transformed vectors T(e1T(e_{1}) and T(e2T(e_{2}).

The transformation provided is defined as: e1e_{1} to e1+2e2e_{1}+2e_{2} and e2e_{2} unchanged. Using the standard basis unit vectors we can define the transformations as:

The matrix of a linear transformation
Equation 18: Transformations of the unit vectors

Notice that T(e2)=e2T(e_{2}) = e_{2} because the transformation is defined to leave the basis unchanged.

3. \quad Draw the transformed vectors using the information provided in the problem.


The matrix of a linear transformation
Figure 3: Drawing the transformed vector in the coordinate plane


4. \quad Combine the the transformed vectors to produce the standard matrix.

The matrix of a linear transformation
Equation 19: Standard matrix A


Example 4

Show that TT is a linear transformation by finding a matrix AA that implements the mapping:

The matrix of a linear transformation
Equation 20: Transformation T(x)T(x) where xx is a 3-dimensional vector

This is the easiest method to use when given the linear transformation definition, the only thing to do is to separate the vector given into its variables' individual vectors:

The matrix of a linear transformation
Equation 21: Expanding the linear transformation into vectors of different variables (part 1)

We can rewrite the transformation above as follows, by factoring out of the column vectors the variables x1x_{1}, x2x_{2} and x3x_{3}:

The matrix of a linear transformation
Equation 22: Expanding the linear transformation into vectors of different variables (part 2)

And so, now we can see how the transformation fits with the formula for a linear transformation T(x)=AxT(x)=Ax by rewriting it once more as the appropriate matrix multiplication of AA and the column vector xx:

The matrix of a linear transformation
Equation 23: Finding the matrix of the linear transformation T(x)=Ax

Therefore, the linear transformation matrix, or standard matrix AA is:

The matrix of a linear transformation
Equation 24: Standard matrix A

***

Now that you have learnt how to compute linear transformations and their corresponding standard matrices, we would like to finalize our lesson be recommending you to take a look into this lesson on linear transformations from RnR^{n} to RmR^{m}, where you can see some more examples and graphs with the transformations explained. We also recommend this lesson named Google search since it goes even deeper on the subject matter by introducing some calculus level computations and linear transformation applications.

So, this is it for our lesson of today, we hope you enjoyed it. Make sure to work through the exercises and see you on the next one!

The matrix of a linear transformation

Lessons

The standard basis, e1e_1 and e2e_2, are unit vectors in R2\Bbb{R}^2 such that:
Standard basis, unit vectors in r^2
If transformed vectors are the transformed vectors, then the standard matrix is
Standard matrix

Why does that work? Watch the intro video

The standard basis, e1e_1, e2e_2, and e3e_3 are unit vectors in R3\Bbb{R}^3 such that:
unit vectors of standard basic e_1, e_2, e_3

If transformed vectors e_1, e_2, e_3 are the transformed vectors, then the standard matrix is
Standard matrix e_1, e_2, e_3

To find the standard basis in R2\Bbb{R}^2 geometrically in a graph we:
1. Draw the standard basis e1e_1 and e2e_2 in the x1x2x_1 x_2 plane
2. Draw the transformed vectors using the information given
3. Identify the transformed vectors T(e1)T(e_1), and T(e2)T(e_2).
4. Combine them to get the standard matrix

Here are the many types of transformations you may see in this section:

Reflections
x1-axis reflection x2-axis reflection
x2=x1 reflection origin reflection

Vertical/Horizontal Expansions and contractions
horizontal expansion/contraction vertical expansion/contraction

Vertical/Horizontal Shears
horizontal shear vertical shear

Projections
projection onto x1 axis projection onto x2 axis

Circle Rotation
circle rotation transformation
  • Introduction
    The Matrix of a Linear Transformation Overview:
    a)
    The Standard Basis and Matrix
    T(x)=Ax,AT(x)=Ax, A: The Standard Matrix
    R2\Bbb{R}^2 standard Basis: r^2 standard basis e1 and r^2 standard basis e2
    R3\Bbb{R}^3 standard Basis: r^3 standard basis e1, e2, e3
    • Transformed standard basis
    • Finding the Standard Matrix

    b)
    Finding the Standard Matrix Geometrically in R2\Bbb{R}^2
    • Drawing the Standard basis on a Graph
    • Identifying the Transformed Standard Basis
    • Combining the Standard Basis' into a Matrix

    c)
    Types of Geometric Linear Transformations in R2\Bbb{R}^2
    • Reflections on a Line
    • Horizontal Expansion/Contraction
    • Vertical Expansion/Contraction
    • Horizontal & Vertical Shears
    • Projections
    • Rotation Transformations


  • 1.
    Finding the Standard Matrix with Transformed Basis
    Assume that TT is a linear transformation. Find the standard matrix of TT if standard matrix of T, T(e1) standard matrix of T, T(e2) where standard matrix of T, e1= [1 0] and standard matrix of T, e2= [0 1].

  • 2.
    Assume that TT is a linear transformation. Find the standard matrix of TT if standard matrix of T, T(e1) standard matrix of T, T(e2), T(e3) where standard matrix of T, e1= [1 0 0], e2= [0 1 0] and standard matrix of T, e3= [0 0 1].

  • 3.
    Finding the Standard Matrix Geometrically
    Assume that TT is a linear transformation. Find the standard matrix of TT if T:R2T: \Bbb{R}^2 R2 \Bbb{R}^2 is a vertical shear transformation that maps e1e_1 to e1+2e2e_1+2e_2 and leaves e2e_2 unchanged.

  • 4.
    Assume that TT is a linear transformation. Find the standard matrix of TT if T:R2T: \Bbb{R}^2 R2 \Bbb{R}^2 first performs a x2=x1x_2=x_1 reflection, and then a x1x_1 reflection.

  • 5.
    Finding the Matrix Algebraically
    Show that TT is a linear transformation by finding a matrix AA that implements the mapping:
    Finding the Matrix Algebraically and show linear transformation

  • 6.
    Finding for x given the Transformation
    Let T:R2T: \Bbb{R}^2 R2 \Bbb{R}^2 be a linear transformation such that Finding for x given the Transformation. Find a xx such that Finding for x given the Transformation