# Section 4.5 $\quad$ Linear Independence

**$\underline{Recall}$** The set $W$ of all vectors of the form
$
  \left[\begin{array}{c} a \\ b \\ a+b \\ \end{array} \right]
$
is a subspace of $\mathbb{R}^3$.

### Example 1

Show that each of the following sets is a spanning set for $W$
\begin{equation*}
  S_1 = \left\{
  \left[\begin{array}{c} 1 \\ 0 \\ 1 \\ \end{array} \right],
  \left[\begin{array}{c} 0 \\ 1 \\ 1 \\ \end{array} \right]
  \right\}, ~~~~~
  S_2 = \left\{
  \left[\begin{array}{c} 1 \\ 0 \\ 1 \\ \end{array} \right],
  \left[\begin{array}{c} 0 \\ 1 \\ 1 \\ \end{array} \right],
  \left[\begin{array}{c} 3 \\ 2 \\ 5 \\ \end{array} \right]
  \right\}
\end{equation*}

## Definition of Linear Dependency

>The vectors  $\mathbf{v}_1, \mathbf{v}_2, \cdots, \mathbf{v}_k$ in a vector space $V$ are said to be **linearly dependent** if <br/><br/><br/><br/>

**Remark**
- $\mathbf{v}_1, \mathbf{v}_2, \cdots, \mathbf{v}_k$ are linearly independent if,<br/><br/><br/><br/>
- $\mathbf{v}_1, \mathbf{v}_2, \cdots, \mathbf{v}_k$ are linearly dependent if,<br/><br/><br/><br/>
- If $S = \{\mathbf{v}_1, \mathbf{v}_2, \cdots, \mathbf{v}_k\}$,<br/><br/><br/><br/>

### Example 2

Determine whether the vectors
\begin{equation*}
  \mathbf{v}_1 = \left[\begin{array}{c} 3 \\ 2 \\ 1 \\ \end{array} \right],~~~~
  \mathbf{v}_2 = \left[\begin{array}{c} 1 \\ 2 \\ 0 \\ \end{array} \right],~~~~
  \mathbf{v}_3 = \left[\begin{array}{c} -1 \\ 2 \\ -1 \\ \end{array} \right],~~~~
\end{equation*}
are linearly independent.


```python
from sympy import *

A = Matrix([[3, 1, -1], [2, 2, 2], [1, 0, -1]]);

A.rank()
```




    2



### Example 3

Are the vectors
\begin{equation*}
  \mathbf{v}_1 = \left[\begin{array}{cc} 2 & 1 \\ 0 & 1 \\ \end{array} \right],~~~~
  \mathbf{v}_2 = \left[\begin{array}{cc} 1 & 2 \\ 1 & 0 \\ \end{array} \right],~~~~
  \mathbf{v}_3 = \left[\begin{array}{cc} 0 & -3 \\ -2 & 1 \\ \end{array} \right],~~~~
\end{equation*}
in $M_{22}$ are linearly independent?


```python
from sympy import *

A = Matrix([[2, 1, 0, 1], [1, 2, 1, 0], [0, -3, -2, 1]]);

A.rank()
```




    2



### Example 4

Determine whether the vectors
\begin{equation*}
  \mathbf{v}_1 = \left[\begin{array}{c} -1 \\ 1 \\ 0 \\ 0 \\\end{array} \right],~~~~
  \mathbf{v}_2 = \left[\begin{array}{c} -2 \\ 0 \\ 1 \\ 1 \\\end{array} \right],
\end{equation*}
are linearly independent.


```python
from sympy import *

A = Matrix([[-1, 1, 0, 0], [-2, 0, 1, 1]]);

A.rank()
```




    2



>**Theorem** Let $S = \{\mathbf{v}_1, \mathbf{v}_2, \cdots, \mathbf{v}_n\}$ be a set of $n$ vectors in $\mathbb{R}^n$.<br/><br/><br/><br/>

### Example 5

Is $S = \{[1~~ 2~~ 3],~~ [0 ~~1~~2],~~ [3~~0~ -1]\}$ a linearly independent set of vectors in $\mathbb{R}^3$?


```python
from sympy import *

A = Matrix([[1, 2, 3], [0, 1, 2], [3, 0, -1]]);

A.rank()
```




    3



>**Theorem** Let $S_1$, $S_2$ be finite subsets of a vector space. Let $S_1$ be a subset of $S_2$. Then
- 
- 

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1. YES
2. YES