# Section 4.6 $\quad$ Basis and Dimension

## Definition of Basis

>The vectors $\mathbf{v}_1, \mathbf{v}_2, \cdots, \mathbf{v}_k$ in a vector space $V$ are said to form a $\underline{basis}$ for $V$ if 
- <br /><br />
- <br /><br />

**Remark**<br /><br /><br /><br />

### Example 1

In $\mathbb{R}^3$, the vectors
$\left[\begin{array}{c} 1 \\ 0 \\ 0 \\ \end{array} \right]$,
$\left[\begin{array}{c} 0 \\ 1 \\ 0 \\ \end{array} \right]$,
$\left[\begin{array}{c} 0 \\ 0 \\ 1 \\ \end{array} \right]$,
form a basis.

**Remark**<br /><br /><br /><br />

### Example 2

Show that the set $S = \{t^2+1,t-1,2t+2\}$ is a basis for the vector space $\mathcal{P}_2$.


```python
from sympy import *

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

A.rank()
```




    3



### Example 3

Find a basis for the subspace $V$ of $\mathcal{P}_2$, consisting of all vectors of the form $at^2+bt+c$, where $c=a-b$.<br /><br /><br /><br /><br /><br /><br />

>**Theorem** If $S = \{\mathbf{v}_1, \mathbf{v}_2, \cdots, \mathbf{v}_k\}$ is a basis for $V$,<br /><br /><br /><br />

**Proof**<br /><br /><br /><br /><br /><br /><br />

>**Theorem** If $S = \{\mathbf{v}_1, \mathbf{v}_2, \cdots, \mathbf{v}_k\}$, and $T = \{\mathbf{w}_1, \mathbf{w}_2, \cdots, \mathbf{w}_l\}$ are bases for $V$, then<br /><br /><br /><br />

### Example 4

Let
\begin{equation*}
  S = \left\{
  \left[\begin{array}{cc} 1 & 1\\ 0 & 0 \\ \end{array} \right],
  \left[\begin{array}{cc} 0 & 0\\ 1 & 1 \\ \end{array} \right],
  \left[\begin{array}{cc} 1 & 0\\ 0 & 1 \\ \end{array} \right],
  \left[\begin{array}{cc} 0 & 1\\ 1 & 1 \\ \end{array} \right]
  \right\},~~~~
  T = \left\{
  \left[\begin{array}{cc} 1 & 1\\ 0 & 0 \\ \end{array} \right],
  \left[\begin{array}{cc} 0 & 0\\ 1 & 1 \\ \end{array} \right],
  \left[\begin{array}{cc} 1 & 0\\ 0 & 1 \\ \end{array} \right]
  \right\}.
\end{equation*}
Is $S$ a basis of $M_{22}$? Is $T$ a basis of $M_{22}$?


```python
from sympy import *

S = Matrix([[1, 1, 0, 0], [0, 0, 1, 1], [1, 0, 0, 1], [0, 1, 1, 1]]);

S.rank()
```




    4



### Example 5

Let $V=\mathbb{R}^3$, $s=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4, \mathbf{v}_5\}$, where $\mathbf{v}_1 = \left[\begin{array}{c} 1 \\ 0 \\ 1 \\ \end{array} \right]$,
$\mathbf{v}_2 = \left[\begin{array}{c} 0 \\ 1 \\ 1 \\ \end{array} \right]$,
$\mathbf{v}_3 = \left[\begin{array}{c} 1 \\ 1 \\ 2 \\ \end{array} \right]$,
$\mathbf{v}_4 = \left[\begin{array}{c} 1 \\ 2 \\ 1 \\ \end{array} \right]$,
$\mathbf{v}_5 = \left[\begin{array}{c} -1 \\ 1 \\ -2 \\ \end{array} \right]$.
Find a subset of $S$ that is a basis for $\mathbb{R}^3$.


```python
from sympy import *

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

A.rank()
```




    3



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1. YES
2. YES