Exercises: Matrices – Serlo

Introduction

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Exercise

Determine the   matrix   whose entries satisfy:
 

Solution

The matrix   is of the form  .

So the result is:  

Exercises on the vector space structure of matrices

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Exercise (Derivation of matrix addition)

Let   be linear maps with

 

Determine the matrices of   with respect to the standard basis. How can you define   such that the result corresponds to the matrix of  ?

In this case, the standard basis corresponds to   with  .

How to get to the proof? (Derivation of matrix addition)

Write the two maps in the same tabular form as used for   above.

You can apply the same method to directly find the matrix representing  .

There is now a very obvious way to define the matrix addition. If you try this, you should get the right result.

Proof (Derivation of matrix addition)

We first determine   by writing down the table and summarizing it as a matrix. For the map   we have

 

This gives us

 

Now we do the same with   to obtain  :

 

We conclude the table into a matrix

 

We are now looking for the matrix of  :

 

This results in the desired matrix

 

We now want to define the addition of two matrices in such a way that  . Remember that we have already defined the vector addition in   component by component - so this definition is a good first attempt. And indeed, with this rule we obtain

 

Solution (Derivation of matrix addition)

If we define the addition of matrices as the addition of the respective components, we arrive at the desired result.

Let   be the above linear map with

 

Exercise (Derivation of scalar multiplication)

Determine the matrix   with respect to the standard basis for the map   and the matrix   for the mapping  . How can you define the multiplication of a matrix by a scalar such that  ?

Solution (Derivation of scalar multiplication)

We already know from the previous exercise that

 

If we now multiply   by the scalar  , we get

 

Whence  . Here you can quickly see that we can also define scalar multiplication element by element. Then

 

Exercises on matrix multiplication

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Exercise (Derivation of matrix multiplication)

Let   be a field and let  . Furthermore, let   and  . Let   be the standard basis of  . Describe   as a function of the entries of   and  .

Solution (Derivation of matrix multiplication)

We already know from the article on matrices of linear maps that   and  . If we write

 

then

 

Now we calculate:

 

Using the same argument as at the beginning of this solution, we now know that

 

Exercise

We are given the matrix  . Calculate the expression  .

Solution

We first consider each summand of the expression to be calculated individually:

 

and bacause

 

we get

 

Together, this results in:

 

Exercise

Use matrix multiplication to prove the addition theorems for the cosine and the sine, i.e.,

 
 

Solution

We consider the rotation matrix   and remember that rotations in the plane can be understood as linear maps. Accordingly, it does not matter whether we rotate directly by an angle  , or first by an angle   and then by  . Therefore, we have

 

A comparison of the entries of the matrices provides the addition theorems to be shown.

Exercises on representative and basis change matrices

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Exercise (Calculating coordinate vectors with respect to a basis)

Let  . Calculate the coordinate vector of   with respect to the basis  .

Solution (Calculating coordinate vectors with respect to a basis)

We wish to find out what the coordinate vector of   looks like with respect to the basis  . This gives us a system of equations that needs to be solved:

 

We now have two equations. On the one hand

 

and on the other

 

Solving this linear system gives   and  . This results in the following for the coordinate vector

 

Exercises on the rank of a matrix

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Exercise

Determine the rank of the following matrix:  

Solution

We transform the matrix   into row-level form and read off the rank of the matrix using the number of zero rows. We obtain:

 

We have created a zero row by converting to row-step form. The rank of our matrix is therefore  .

In this case, the shorthand   indicates that we have added the third row of the matrix with   times the second row

Exercise

Determine the rank of the following matrix:  

Solution

We transform the matrix   into row-level form and read off the rank of the matrix using the number of zero rows. We obtain:

 

By converting to row-level form, we have thus shown that for   we have  .

However, we could have seen much more quickly at this point that  . It is also sufficient to show that the column vectors (or equivalently the row vectors) are linearly independent. In this example, we decide to use the column vectors and show their linear independence. Let  .

 

This gives us the linear system:

 

with the unique solution  , which shows the linear independence of the column vectors. However, the rank of a matrix describes the maximum number of linearly independent column vectors of the matrix. So  .

The exercise therefore shows that it is not always advantageous to convert the matrix into row-level form in order to read off the rank of the matrix.

Exercises on matrix inversion

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Exercise

Let   be invertible. Furthermore, let  . Show that   is self-inverse, i.e.,  .

Solution

Since   is invertible, there exists an   with  . Thus, we can write