Linear Programming Lecture Notes - Penn State Personal Web Server

Gauss-Jordan elimination to solve this matrix equation yielding: α1 = α2 = α3 = 0. Thus
these vectors are linearly independent.
Remark 3.31. It is worthwhile to note that the zero vector 0 makes any set of vectors
a linearly dependent set.
Exercise 32. Prove the remark above.
Exercise 33. Show that the vectors
⎡
x1 = ⎣ 1
⎤ ⎡
2⎦
, x2 = ⎣
3
4
⎤ ⎡
5⎦
, x3 = ⎣
6
7
⎤
8⎦
9
are not linearly independent. [Hint: Following the example, create a matrix whose columns
are the vectors in question and solve a matrix equation with right-hand-side equal to zero.
Using Gauss-Jordan elimination, show that a zero row results and thus find the infinite set
of values solving the system.]
Remark 3.32. So far we have only given examples and exercises in which the number
of vectors was equal to the dimension of the space they occupied. Clearly, we could have,
for example, 3 linearly independent vectors in 4 dimensional space. We illustrate this case
in the following example.
Example 3.33. Consider the vectors:
⎡
x1 = ⎣ 1
⎤ ⎡
2⎦
, x2 = ⎣
3
4
⎤
5⎦
6
Determining linear independence requires us to solve the matrix equation:
⎡ ⎤ ⎡
1 4
α1 ⎣2 5⎦
= ⎣
α2
3 6
0
⎤
0⎦
0
The augmented matrix:
⎡ ⎤
1 4 0
⎣ 2 5 0 ⎦
4 6 0
represents the matrix equation. Using Gauss-Jordan elimination yields:
⎡ ⎤
1 4 0
⎣ 0 1 0 ⎦
0 0 0
This implies that the following system of equations:
α1 + 4α2 = 0
α2 = 0
0α1 + 0α2 = 0
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