LAWSON DONG / FIELD NOTES
MATHEMATICS / VISUAL NOTES

Linear Algebra.

This corner of the notebook is mainly for interactive visualizations. Move through the operations and watch the matrix, equations, and geometry describe the same system.

SHARED EXAMPLE · Ax=bAx=b
R1
R2

Change any of the six values; both visualizations update together.

01 / VISUALIZATION

Gaussian Elimination.

Step through a purposeful sequence of EROs. A row changes, a line moves, and the common intersection stays fixed.

CURRENT OPERATION

Start

Two rows, two constraints. Choose a nonzero pivot in the first column.

AUGMENTED MATRIX[1132−10]\left[\begin{array}{cc|c}1 & 1 & 3\\2 & -1 & 0\end{array}\right]
1x+1y=31x + 1y = 32x−1y=02x - 1y = 0
STEP 1 / 4
GEOMETRY AT THIS STEPSolution (1, 2)
xy(1, 2)
Row 1 Row 2● Common solution

The step that clears below the first pivot is Gaussian elimination. Clearing above a pivot afterward is the Gauss–Jordan extension.

02 / VISUALIZATION

Elementary Row Operations.

Explore the three moves independently. Gaussian elimination uses these same moves, choosing multipliers that make selected coefficients zero.

CURRENT MOVE

R2←R2+cR1R_2 \leftarrow R_2 + cR_1

Add a multiple of row 1 to row 2. Move c until a coefficient becomes zero.

−404
RESULTING MATRIX[1130−3−6]\left[\begin{array}{cc|c}1 & 1 & 3\\0 & -3 & -6\end{array}\right]
1x+1y=31x + 1y = 30x−3y=−60x - 3y = -6
GEOMETRY OF THE NEW ROWSSolution (1, 2)
xy(1, 2)
Row 1 Row 2● Common solution
03 / VISUALIZATION

Column space & consistency.

Edit the augmented matrix [A | b]. Watch the column combinations in output space and the two row equations in input space describe the same system.

b is a linear combination of columns of A  ⟺  Ax=b\iff Ax=b has a solution  ⟺  Ax=b\iff Ax=b is consistent
[A∣b][A\mid b]
Consistentb is a linear combination of the columns of Arank⁡(A)=1,Col⁡(A)=a line through the origin\operatorname{rank}(A)=1,\quad\operatorname{Col}(A)=\text{a line through the origin}

The columns span one line. Ax always stays on this line; b is reachable exactly when it lies on the line.

Row picture · equations in input space

Each row gives a line in the (x₁, x₂) plane. A shared point solves both equations. Drag x or adjust x₁ and x₂ below to explore column combinations.

ConsistentThe lines coincide: every point on their shared line is a solution.

R1:1x1+2x2=3R_1: 1x_1 + 2x_2 = 3

R2:1x1+2x2=3R_2: 1x_1 + 2x_2 = 3

Left-click and drag x to change the column-combination coefficientsxx₁x₂-5.405.4
R1 · solid lineR2 · dashed linex · draggable coefficients

Each row of [A | b] defines an equation in the input plane. Its coefficients form a normal vector to the line; these equation lines are not the subspace Row(A).

Output space · Ax and b

Drag the red b handle to test which targets are reachable.

a₁a₂Ax = bx₁a₁x₂a₂Drag b06-66

Scroll over this canvas to zoom, or use + / −. Zoom is centered on the origin. Auto-fit follows x.

a₁a₂x₁a₁x₂a₂ · starts at the tip of x₁a₁AxbShading · Col(A)
Current coefficients: x1=1,x2=1x_1 = 1,\quad x_2 = 1

1a1+(1)a2=(3,3)=b1a_1 + (1)a_2 = (3, 3) = b

These coefficients reach b.

Rank and line membership use relative numerical tolerance 10⁻¹⁰. Extremely close cases are treated as equal at this precision.

HOW THEY FIT TOGETHER

The moves and the strategy.

Each row of an augmented matrix is an equation. Swapping rows changes their order; scaling a row by a nonzero number rewrites the same equation; replacing a row by itself plus a multiple of another produces an equivalent system.

Gaussian elimination uses those three moves with a particular aim: choose a pivot and make the entries beneath it zero. Gauss–Jordan elimination continues by clearing above the pivots.

Allowed movesElementary row operationsStrategy for creating zerosGaussian elimination