6.1-06 The complete four-subspace picture

The complete four-subspace picture

Bring the four fundamental subspaces into one coherent account of a linear map. This questionset emphasizes which spaces belong to the domain, which belong to the codomain, and how rank and orthogonality connect them.

What you will practise

• Locate and count all four subspaces.
• Explain what happens to a null-space component.
• Describe the map from row space to column space.
• Interpret minimum-norm solutions through perpendicular components.

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 the row–null and column–left-null relationships. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports least squares, inverse problems and later singular-value analysis within Module 6 of the Complete Linear Algebra Course.