š Row Picture (Lines)
When you click "Solve System", you see the row picture:
- Each equation represents a line in 2D space
- $2x + y = 8$ is line $L_1$
- $4x - 3y = 6$ is line $L_2$
- The solution is where the lines intersect
This is how we traditionally visualize linear systems.
š Column Picture (Vectors)
When you click "Show Linear Combination", you see the column picture:
- The system becomes: $c_1 \mathbf{v_1} + c_2 \mathbf{v_2} = \mathbf{b}$
- $\mathbf{v_1}$ and $\mathbf{v_2}$ are the column vectors
- We find scalars $c_1, c_2$ that combine them to reach $\mathbf{b}$
- The parallelogram shows the vector addition
This reveals the geometric structure of solutions.
š How They Connect
The inputs are synchronized! For the system:
$$\begin{cases}
2x + y = 8 \\
4x - 3y = 6
\end{cases}$$
Row form: Two equations, two unknowns
Column form:
$$x \begin{bmatrix} 2 \\ 4 \end{bmatrix} + y \begin{bmatrix} 1 \\ -3 \end{bmatrix} = \begin{bmatrix} 8 \\ 6 \end{bmatrix}$$
The column vectors are: $\mathbf{v_1} = \begin{bmatrix} 2 \\ 4 \end{bmatrix}$ and $\mathbf{v_2} = \begin{bmatrix} 1 \\ -3 \end{bmatrix}$
The solution $x=3, y=2$ becomes $c_1=3, c_2=2$ in the vector view!
āØ Change either the equations or vectors - the other updates automatically!