Review of Some Linear Algebra

In this discussion, we expect some familiarity with matrices.  For a review of the basics click here.  We will rely heavily on calculators and computers to work out the problems.  Consider some examples.  

 

Example

Solve the system of equations

        4x + y + 3z  =  2
        x - 2y - 5z  =  3
        5x + 2z  =  1

Solution

We write this system as the matrix equation 

        Ax  =  b

where 

                       

To solve this matrix equation we take the inverse of both sides (which is possible since the determinant of A is -13 which is not equal to zero.  We have 

        x  =  A-1b

Using a calculator we find that 

       

Multiplying by b gives

       

What we mean by "x" is the vector <x,y,z>.  The solution is 

        x  =  1        y  =  4        z  =  -2

 

Example

Find the solution of 

        3x + 2y - z  =  5
        2x + y - z  =   2
        5x + 4y - z  =  11

Solution

A quick check shows that we cannot solve this problem in the same way, since the determinant of A is 0.  Instead, we rref the augmented matrix 

       

to get

       

Putting this back into equation form, we get

        x - z  =  -1     and       y + z  =  4  

We write this as

        x  =  -1 + z        y  =  4 - z        z  =  z

Letting z  =  t be the parameter we get parametric equations for the solution set

          x  =  -1 + t        y  =  4 - t        z  =  t   

   

Recall that vectors v1, ..., vn are called linearly independent if

        c1v1 + ... + c2v2  =  0 

implies that all of the constants ci are zero.  A theorem from linear algebra tell us that if we have n vectors in Rn then they will be linearly independent if and only if the determinant of the matrix whose columns are these vectors has nonzero determinant.

 

Example

Show that the vectors 

        u  =  <1,4,-2>    v  =  <0,3,5>    and    w  =  <1,2,3> 

are linearly independent.

Solution

We find the determinant

       

Since the determinant is nonzero, the vectors are linearly independent.

 

For systems of differential equations, eigenvalues and eigenvectors play a crucial role.  We recall their definitions below

Definition:   Eigenvalues and Eigenvectors

Let A be an n x n matrix.  Then l is an eigenvalue for A with eigenvector v if 

          Av  =  lv

 

Example 

Find the eigenvalues and eigenvectors for 

       

Solution

If 

        Av  =  lv

then

        A - lI  =  0 

Taking determinants of both sides, we get

        (6 - l)(-1 - l) + 12  =  0

        l2 - 5l + 6  =  0

        (l - 2)(l - 3)  =  0

The eigenvalues are 

        l  =  2     and        l  =  3

To find the eigenvectors, we plug the eigenvalues into the equation 

           A - lI  =  0 

and find the null space of the left hand side.  For the eigenvalue l  =  2, we have

         

The first row gives

        y  =  -x

so that an eigenvector corresponding to the eigenvalue l  =  2 is 

       

For the eigenvalue l  =  3, we have

         

The first row gives

        3y  =  -4x

so that an eigenvector corresponding to the eigenvalue l  =  3 is 

       

Typically, we want to normalize the eigenvectors, that is find unit eigenvectors.  We get

       

 


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