Linear Algebra

模块目标

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

课程大纲

1. 01 Problem and prerequisites 15h

   学习结果 Restate solvability in the language of span and linear combination.

   形成性检查 四选一检查

2. 02 Vectors and linear combinations 20h

   学习结果 Decide whether a target vector lies in a stated span and justify the decision.

   形成性检查 书面推导

3. 03 Rank and solvability of linear systems 20h

   学习结果 Relate rank of A and of the augmented matrix to existence of solutions.

   形成性检查 四选一检查

4. 04 Geometric foundations: orthogonality 20h

   学习结果 Use an inner product to test orthogonality and to split a vector into parallel and perpendicular parts.

   形成性检查 书面推导

5. 05 Orthogonal projection 20h

   学习结果 Derive the projection formula onto a subspace with an orthogonal or orthonormal generating set.

   形成性检查 书面推导

6. 06 Derivation: normal equations 25h

   学习结果 Derive AᵀAx = Aᵀb from minimising ‖Ax−b‖² and state when the minimiser is unique.

   形成性检查 书面推导

7. 07 Uniqueness and conditioning 15h

   学习结果 Give a rank condition for uniqueness and a counterexample where a small change in b changes x sharply.

   形成性检查 反例

8. 08 Methods and examination synthesis 15h

   学习结果 Assemble a complete least-squares argument: existence, geometry, algebra, uniqueness.

   形成性检查 书面推导

学习结果

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.

学科技能

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

可迁移技能

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

相关入门主题

01Problem and prerequisites2 个主题

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

章节链接

02Geometric foundations2 个主题

1. Orthogonality and residuals
2. Orthogonal projection

章节链接

03Derivation and solution2 个主题

1. Normal equations
2. Existence and uniqueness

章节链接

04Methods and examination synthesis2 个主题

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

章节链接