# Linear independence

### Linear independence

#### Lessons

We say that a set of vectors {$v_1, \cdots , v_p$} in $\Bbb{R}^n$ is linearly independent if:
$v_1 x_1+v_2 x_2+\cdots+v_p x_p=0$

gives only the trivial solution. In other words, the only solution is:

We say that a set of vectors {$v_1, \cdots , v_p$} in $\Bbb{R}^n$ is linearly dependent if:
$v_1 x_1+v_2 x_2+\cdots+v_p x_p=0$

gives a non-trivial solution. In other words, they are linearly dependent if it has a general solution (aka has free variable).

We can determine if the vectors is linearly independent by combining all the columns in matrix (denoted as A) and solving for

$Ax=0$

Fast way to tell if 2 or more vector are linearly dependent
1. The vectors are multiples of one another
2. There are more vectors than there are entries in each vector.
3. There is a zero vector
• 1.
Linear Independence Overview:
a)
Linearly independent
• Definition of linear independence
• Trivial solution

b)
Linearly dependent
• Definition of linear dependence
• Non-trivial solutions

c)
Fast ways to determine linear dependence
• Vectors are multiples of one another
• # of Vectors > # of Entries in each vector
• Zero Matrix

• 2.
Determining Linear independence by solving
Determine if the following vectors are linearly independent:

• 3.
Determine if the matrix is linearly independent by solving $Ax=0$:

• 4.
Determining Linear dependence by inspection
Determine by inspection if the following vectors are linearly dependent:
a)

b)

c)

d)

• 5.
Linear dependence/independence with unknown constant
Find the value(s) of $k$ for which the vectors are linearly dependent.