Elimination and Subspace Diagnostics Lab

Retrieval Prompts

1. State the three legal row operations from memory.
2. State what a pivot column tells you.
3. State the difference between Ax = 0 and Ax = b.
4. State what it means for a set to be a subspace.
5. State what the column space and nullspace each represent.

Compare and Distinguish

Separate these pairs clearly:

• contradictory row versus free-variable row
• subspace versus affine set
• column space versus nullspace
• row-equivalent matrix versus same column space

Common Mistake Check

Identify the mistake in each claim:

1. "If two systems look different after elimination, they have different solutions."
2. "Any plane in R^3 is a subspace."
3. "The pivot columns of the reduced matrix are a basis for the original column space."
4. "If a homogeneous system has one free variable, it still only has the zero solution."

Mini Application

For one 3 x 4 matrix of your choice:

1. row-reduce it
2. find the rank
3. describe the nullspace parametrically
4. name a basis for the column space
5. explain in one paragraph what information the matrix preserves and what information it destroys

Evidence Check

This page is complete only if you can inspect a reduced matrix and immediately say:

• whether Ax = b is consistent
• how many free variables exist
• what the rank is
• what subspace the outputs live in