1$\text{How do you find the basis for a } \text{span} \{v_1, v_2, \dots\}?$
$1. \quad \text{Create } A = [v_1 \text{ } v_2 \dots]$
$2. \quad \text{Find RREF}(A)$
$3. \quad \text{Identify columns of RREF}(A) \text{ that have pivots (e.g., column 1, column 2, \dots)}$
$4. \quad \text{Basis} = \{\text{column 1 of } A, \text{column 2 of } A, \dots\}$
2$\text{What is the condition for } \mathcal{A} \text{ to be a spanning set of } V?$
$\text{span}(\mathcal{A}) = V$
3What is the method to find the spanning set of V = {v_1, v_2 ...}?
$1. \quad \text{Flatten and combine } v_1, v_2, \dots \text{ into a matrix } A$
$2. \quad \text{Find RREF}(A)$
$3. \quad \text{Identify columns of RREF}(A) \text{ that have pivots (e.g., column 1, column 2, \dots)}$
$4. \quad = \text{Span} \{v_1, v_2, \dots\}$
4$\text{What is the method to find the spanning set of } V = \left\{ \begin{bmatrix} a & b \\ c & d \end{bmatrix} : \text{conditions} \right\}?$
$1. \quad \text{A general element of } V = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$
$2. \quad \begin{bmatrix} a & b \\ c & d \end{bmatrix} = a \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} + b \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} + c \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} + d \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}$
$3. \quad = \text{Span}\left\{ \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \right\}$
5$\text{What is the method to check if } \{v_1, v_2, v_3, v_4, \dots\} \text{ spans } M_{m \times n}(\mathbb{F})?$
$1. \quad \text{If the set has } (m \times n) \text{ elements, continue. If not, it doesn't span.}$
$2. \quad \text{Flatten and combine } v_1, v_2, v_3, v_4... \text{ into a matrix } A$
$3. \quad \text{If } A \text{ is right invertible, the set spans } M_{m \times n}(\mathbb{F}), \text{ if not, it doesn't}$
8$\text{How do you flatten }\begin{bmatrix} a & b \\ c & d \end{bmatrix}?$
$\begin{bmatrix} a & b \\ c & d \end{bmatrix} \to \begin{bmatrix} a \\ b \\ c \\ d \end{bmatrix}$