Dear linear algebra students, This is what matrices (and matrix manipulation) really look like

3-Second Blueprint (English)

1. Linear systems map geometrically either to intersecting hyperplanes (row view) or to scaling column vectors to reach a target vector.
2. Because Ax = 0 enforces zero dot products with every row vector, the matrix row space is strictly orthogonal to its null space.
3. Gaussian elimination rotates hyperplanes around their intersection; in circuit node matrices, the null space defines zero-current states.