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
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
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.