6.3-01 A line projection as a matrix

A line projection as a matrix

Projection onto a line is the smallest complete model of orthogonal approximation. Work from a scalar orthogonality equation to the outer-product matrix, then check scaling invariance, signs, distances and the range of the result.

What you will practise

• Derive the projection coefficient.
• Construct and apply line-projector matrices.
• Separate projection from residual.
• Use squared lengths to calculate distance.

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 dot products, norms and outer products. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports projection onto higher-dimensional subspaces within Module 6 of the Complete Linear Algebra Course.