6.1-04 Column space and left null space

Column space and left null space

Column space and left null space describe complementary parts of the output space. Practice moves from the transpose equation to concrete reachability tests and certificates that a target cannot be produced by a matrix.

What you will practise

• Identify the left null space and its ambient dimension.
• Compute perpendicular output directions.
• Test targets against compatibility conditions.
• Explain why a left-null vector can certify inconsistency.

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 transpose multiplication, column spans and homogeneous systems. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports understanding the location of least-squares residuals within Module 6 of the Complete Linear Algebra Course.