Linear Algebra

Module aims

State when a linear system is solvable, relate projection to the normal equations, and distinguish uniqueness from numerical sensitivity.

Course outline

1. 01 Problem and prerequisites 15h

   Learning result Restate solvability in the language of span and linear combination.

   Formative check Four-option check

2. 02 Vectors and linear combinations 20h

   Learning result Decide whether a target vector lies in a stated span and justify the decision.

   Formative check Written derivation

3. 03 Rank and solvability of linear systems 20h

   Learning result Relate rank of A and of the augmented matrix to existence of solutions.

   Formative check Four-option check

4. 04 Geometric foundations: orthogonality 20h

   Learning result Use an inner product to test orthogonality and to split a vector into parallel and perpendicular parts.

   Formative check Written derivation

5. 05 Orthogonal projection 20h

   Learning result Derive the projection formula onto a subspace with an orthogonal or orthonormal generating set.

   Formative check Written derivation

6. 06 Derivation: normal equations 25h

   Learning result Derive AᵀAx = Aᵀb from minimising ‖Ax−b‖² and state when the minimiser is unique.

   Formative check Written derivation

7. 07 Uniqueness and conditioning 15h

   Learning result Give a rank condition for uniqueness and a counterexample where a small change in b changes x sharply.

   Formative check Counterexample

8. 08 Methods and examination synthesis 15h

   Learning result Assemble a complete least-squares argument: existence, geometry, algebra, uniqueness.

   Formative check Written derivation

Learning results

Given a concrete system or least-squares problem, produce a derivation that names rank, column space, and the normal equations, and say what uniqueness does and does not follow.

Subject skills

Row reduction, rank, orthogonality, orthogonal projection, normal equations, residual geometry.

Transferable skills

Separate geometric claims from algebraic identities; write a short derivation that a reader can check line by line.

Related introductory topics

01Problem and prerequisites2 topics

1. Solvability of linear systems
2. Column space, rank and nullity

Chapter link

02Geometric foundations2 topics

1. Orthogonality and residuals
2. Orthogonal projection

Chapter link

03Derivation and solution2 topics

1. Normal equations
2. Existence and uniqueness

Chapter link

04Methods and examination synthesis2 topics

1. Conditioning and method selection
2. Least-squares examination synthesis

Chapter link