<jupyter_start><jupyter_text>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 - -  **Remark**  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**  Example 2 Show that the set $S = \{t^2+1,t-1,2t+2\}$ is a basis for the vector space $\mathcal{P}_2$.<jupyter_code>from sympy import *

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

A.rank()<jupyter_output><empty_output><jupyter_text>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 $c=a-b$. >**Theorem** If $S = \{\mathbf{v}_1, \mathbf{v}_2, \cdots, \mathbf{v}_k\}$ is a basis for $V$, **Proof** >**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  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 $M_{22}$?<jupyter_code>from sympy import *

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

S.rank()<jupyter_output><empty_output><jupyter_text>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 $\mathbb{R}^3$.<jupyter_code>from sympy import *

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

A.rank()<jupyter_output><empty_output>
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