Review of Some Linear Algebra

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, ..., v_n are called linearly independent if

c₁v₁ + ... + c₂v₂ = 0

implies that all of the constants c_i 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