Sample notes
The Geometry of Linear Equations
Source: The Geometry of Linear Equations · MIT OpenCourseWare · Gilbert Strang · 47-second excerpt · English audio
Unknowns and the right-hand side
Lecture audio · 00:00–00:11- Key idea: The unknowns x and y form one vector; the right-hand-side values form a different vector.
- Worked example: A = [[2, −1], [−1, 2]]; A · (x, y)ᵀ = (0, 3)ᵀ.
- Check yourself: Which vector contains the right-hand-side values?
Check both equations
Lecture audio · 00:11–00:23- Key idea: Substitute the values into every equation. A solution makes both sides equal.
- Worked example: For x = 1 and y = 2: 2×1 − 2 = 0, and −1 + 2×2 = 3. Both equations hold.
- Check yourself: Does (x, y) = (1, 2) solve this two-equation system?
The x column in a three-variable system
Lecture audio · 00:35–00:47- Key idea: In the later three-variable example, collect the coefficient of x from each equation in row order.
- Worked example: The x coefficients are 2, −1, 0, so its column is (2, −1, 0)ᵀ. Zero means x is absent from that equation.
- Check yourself: What is the x column in the later three-variable example?
