Lab 3.1
Finding the Null Space
Lab goal
Identify pivot and free variables, then construct and verify one null-space basis vector.
- Null space
- Pivot and free variables
- Rank and nullity
- Linear independence
Code extensionThe full notebook detects pivots in a random 10×11 matrix and constructs several null-space basis vectors at once.
Key ideas
Terms you will use
- Null space
Null(A) = {x : Ax = 0} The null space contains every input vector that A sends to the zero output.
- Free variable
pivot part = −Uₚ⁻¹U𝒻 · free part A non-pivot variable can be chosen first; the pivot variables then adjust to satisfy the equations.
- Rank–nullity
rank(A) + nullity(A) = n The input coordinates split between pivot directions and null directions.
Lab 3.1 · Guided example
Build a direction that disappears
The notebook’s random 10×11 example is reduced to one 2×3 matrix so every pivot, free variable, and cancellation stays visible.
Select a line to see its action, array shape, and concrete operation.
Which of the three columns will become free?
Three input coordinates, two output coordinates
Because A maps R³ into R², we look for an input direction that collapses to zero.
Reveal to compare.
A.shapenot stored → ?Three-dimensional inputs, two-dimensional outputs.