1. The Machine Setup: Matrix $A$ is on the left, Vector $\mathbf{x}$ is on the right.
2. Row-by-Vector: Multiply each row of the matrix by the vector using the Dot Product.
3. Sum the Products: Each row produces one entry in the result vector $\mathbf{b}$.
4. Dimension Check: The number of columns in $A$ must equal the number of entries in $\mathbf{x}$.
Calculate the output vector $\mathbf{b} = A\mathbf{x}$ for each scenario.
$A = \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix}$ and $\mathbf{x} = \begin{bmatrix} 5 \\ 4 \end{bmatrix}$
$A = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}$ and $\mathbf{x} = \begin{bmatrix} 10 \\ 2 \end{bmatrix}$
$A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ and $\mathbf{x} = \begin{bmatrix} 7 \\ -3 \end{bmatrix}$
Note: This is a Reflection Matrix!
Give the dimension of the Domain (Input) and the Codomain (Output) for each Matrix.
$A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$
$B = \begin{bmatrix} 10 \\ 20 \\ 30 \end{bmatrix}$
If you double the input vector ($2\mathbf{x}$), what happens to the output vector? Does the machine $A$ also double the result? Test it using $A = \begin{bmatrix} 2 & 1 \\ 0 & 1 \end{bmatrix}$ and $\mathbf{x} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$.
_________________________________________________________________________
_________________________________________________________________________
The Sacrifice: A matrix $A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$ represents a process of "Equalization."
Find an input vector $\mathbf{x}$ (not zero!) that this matrix maps to zero ($A\mathbf{x} = \mathbf{0}$).
Hint: What if $x$ and $y$ are opposites?
A person passes through two ministries.
Ministry 1 ($A$) = $\begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}$ (Multiplies X by 2).
Ministry 2 ($B$) = $\begin{bmatrix} 1 & 0 \\ 0 & 5 \end{bmatrix}$ (Multiplies Y by 5).
Task: If the person starts with $\mathbf{x} = \langle 1, 1 \rangle$, find the final result after both transformations ($B(A\mathbf{x})$).
Objective: Explain "Transform" to a younger student using their hand.
The Activity:
1. Have them point their finger North ($x$).
2. Say "Matrix Command: Turn 90 degrees."
3. They point East ($b$).
The Lesson: "In math, we use Grids to give commands to arrows. The arrow doesn't move itself; the Grid moves it!"
Response: __________________________________________________________