1$\text{What are the 2 steps to check if a transformation } T: V \to W \text{ is a linear transformation?}$
$1. \quad \text{Create a general } v \in V \text{ and } w \in W$
$2. \quad \text{Check } T(v + w) = T(v) + T(w)$
$3. \quad \text{Check } T(\alpha v) = \alpha T(v)$
2$\text{What is a property of the identity transformation } T: V \to V?$
$T(v) = v$
3$\text{What is a property of the zero transformation } T: V \to V?$
$T(v) = \underline{0}$
4$\text{What is the method to find } T_A, \text{given T(B)=C?}$
$1. \quad \text{Find } T(e_1), T(e_2), \dots, T(e_n)$
$2. \quad T_A = [T(e_1) \, T(e_2) \, \dots \, T(e_n)]$
5$\text{What is the method to find } T(c) \text{ given: } T: V \to W, \, \beta = \{b_1, b_2, \dots\}?$
$1. \quad \text{Verify that } \beta \text{ is a basis of } V$
$2. \quad \text{Flatten and combine } b_1, b_2, \dots \text{ into matrix } A$
$3. \quad \text{Solve } A x = C \, (\text{where } c \text{ is a general matrix})$
$4. \quad \text{Unflatten the adjusted } C \text{ if } b_1, b_2, \dots \text{ weren't flat}$
$5. \quad T(C) = x_1 T(b_1) + x_2 T(b_2) + \dots$
$6. \quad \text{Simplify RHS for } T(c) = \dots$
6Method to find T(c) given T_A..
Is this possible
7$\text{What is the method to find the matrix representation of } T \circ S : \mathbb{F}^p \to \mathbb{F}^n$
$1. \quad \text{Find } T_A = A$
$2. \quad \text{Find } S_B = B$
$2. \quad T_A \circ S_B : v \mapsto (AB)v$ = $(T_A \circ T_B)v= (AB)v$
8$\text{If } T: V \to W \text{ is bijective, what can we say?}$
$T^{-1} \text{ is also a linear transformation}$
9$\text{What is the method to find the basis } (\beta) \text{ of } \ker(T), \text{ where } T \text{ is given by } T(A) = B?$
$1. \quad \text{Solve } B = \underline{0} \text{ for } x_1, x_2, \dots$
$2. \quad B = x_1 [\dots] + x_2 [\dots] + \dots$
$3. \quad \ker(T) = \text{span}([\dots], [\dots], \dots)$
$4. \quad \text{Find } \beta \text{ from } = \text{span}\{[\dots], [\dots], \dots\}$
10$\text{What is the method to find the basis } (\beta) \text{ of } \ker(T_A)?$
$1. \quad \text{Find the right null space of } T_A \text{ in the form } = \text{span}\{[], [], \dots\}$
$2. \quad \ker(T) = \text{span}\{[], [], \dots\}$
$3. \quad \text{Find } \beta \text{ from } = \text{span}\{[\dots], [\dots], \dots\}$
11$\text{What is the method to find the basis } (\beta) \text{ of } \text{im}(T), \text{ where } T \text{ is given by } T(A) = B?$
$1. \quad \text{Write } B = x_1 [\dots] + x_2 [\dots] + \dots$
$2. \quad \text{im}(T) = \text{span}([\dots], [\dots], \dots)$
$3. \quad \text{Find } \beta \text{ from } = \text{span}\{[], [], \dots\}$
12$\text{What is the method to find the basis } (\beta) \text{ of } \text{im}(T_A)?$
$1. \quad \text{Find the column space of } T_A \text{ in the form } \text{span}\{[], [], \dots\}$
$2. \quad \text{im}(T_A) = \text{span}\{[], [], \dots\}$
$3. \quad \text{Find } \beta \text{ from } = \text{span}\{[], [], \dots\}$
13What is rank(T) = ?
dim(imT)
14What is null(T) = ?
dim(kerT)
15$\text{How is } \text{rank}(T) \text{ related to } \text{null}(T) \text{ for } T: V \to W?$
rank(T) + null(T) = n
16Image of $T, T_A$ is the?
The column space of $T, T_A$
17kernel of $T, T_A$ is the?
The right null space of $T, T_A$