6.1-03 Row space and null space

Row space and null space

Explore the relationship between the null space and row space of a real matrix. This set develops arguments from row-by-vector multiplication, then uses them to interpret kernels, rank and the action of a matrix on its domain.

What you will practise

• Read a zero output as row orthogonality.
• Prove both directions of the row–null relationship.
• Recover null directions from row-reduced equations.
• Explain injectivity on the row space.

Begin with recognition and interpretation, then work through calculations, error diagnosis and more demanding reasoning. Single-select and multi-select questions ask you to evaluate the stated assumptions as well as the result. Worked explanations connect the answer to a method or a counterexample, so you can check why tempting alternatives fail.

Before you start

Review row reduction, rank–nullity and matrix–vector products. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports the input-side half of the complete four-subspace picture within Module 6 of the Complete Linear Algebra Course.