1What are the 8 vector space axioms?
$
\begin{array}{rl}
\text{(V1)} & u + v = v + u \quad \forall \, u, v \in V \\
\text{(V2)} & (u + v) + w = u + (v + w) \quad \forall \, u, v, w \in V \\
\text{(V3)} & \exists \, \mathbf{0} \in V \text{ such that } v + \mathbf{0} = v \quad \forall \, v \in V \\
\text{(V4)} & \forall \, v \in V, \, \exists \, u \in V \text{ such that } v + u = \mathbf{0} \\
\text{(V5)} & \alpha (u + v) = \alpha u + \alpha v \quad \forall \, u, v \in V \text{ and all scalars } \alpha \\
\text{(V6)} & (\alpha + \beta) v = \alpha v + \beta v \quad \forall \, v \in V \text{ and all scalars } \alpha, \beta \\
\text{(V7)} & (\alpha \beta) v = \alpha (\beta v) \quad \forall \, v \in V \text{ and all scalars } \alpha, \beta \\
\text{(V8)} & 1 \cdot v = v \quad \forall \, v \in V \quad (\text{where } 1 \in F)
\end{array}
$
2What are the two operations of vector spaces?
$
\alpha \in \mathbb{F} \, * \, v \in \mathbb{V} \implies \alpha v \in \mathbb{V}
$
$
v \in V + u \in V \implies (u + v) \in V
$
3What are the 4 steps to check if a vector space W is a subspace of a vector space V?
$
1. \quad \text{Check } \underline{0} \in W
$
$
2. \quad \text{Create general } v \in V \text{ and } w \in W
$
$
3. \quad \text{Check } v + w \in W
$
$
4. \quad \text{Check } \alpha v \in W
$
4$\text{What are the steps to find the right null space of } A \in M_{m \times n} (\mathbb{F})?$
$1. \quad \text{Solve } A x = 0 \text{ for } x = x_1() + x_2() + \dots$
$2. \quad \text{Right null space of } A = \text{span} \{ (), (), \dots \}$
5$\text{What are the steps to find the column space of } A \in M_{m \times n} (\mathbb{F})?$
1. Find RREF(A)
2. I dentify columns with pivots, columns 1,2 ...
3. Column space of A = span{column 1 of A, column 2 of A...}
6$\text{What is the left null space of } A \in M_{m \times n} (\mathbb{F})?$
$\text{Right null space of } A^T$
7$\text{What is the row space of } A \in M_{m \times n} (\mathbb{F})?$
$\text{Column space of } A^T$
8$\text{What is the method to find a left invertible matrix } N \in M_{n \times (n-r)} (\mathbb{F}) \text{ such that the column space of } N = \text{right null space of } A \in M_{m \times n} (\mathbb{F})?$
$1. \quad \text{Find right null space of } A, = \text{span} \{ (), (), \dots \}$
$2. \quad \text{Find a basis for the right null space of } A$
$3. \quad \text{The vectors in the basis are the columns of } N$
9$\text{What is the method to find a right invertible matrix } N \in M_{m \times r} (\mathbb{F}) \text{ such that the column space of } N = \text{column space of } A \in M_{m \times n} (\mathbb{F})?$
1. Find RREF(A)
2. Create a set containing the columns in RREF(A) that have pivots
3. The vectors in the set are the columns of N
10$\text{What is the method to find a right invertible matrix } N \in M_{(m-r) \times m} (\mathbb{F}) \text{ such that the right null space of } N = \text{column space of } A \in M_{m \times n} (\mathbb{F})?$
1. Find RREF(A)
2. Create a set containing the rows in RREF(A) that do not have pivots
3. The vectors in the set are the rows of N
11$\text{If } A \subseteq B \text{ and } B \subseteq C, \text{ is } A \subseteq C?$
Yes
12$\text{If } A \text{ is a subspace of } C \text{ and } B \text{ is a subspace of } C, \text{ what must be true for } A \text{ to be a subspace of } B?$
A in B