Linear algebra, TCD 2014/15
MA1111
- Linear algebra in 2d and 3d. Vectors. Dot and cross products.
- Systems of simultaneous linear equations. Gauss--Jordan elimination.
- (Reduced) row echelon form for a rectangular matrix. Principal and free variables. Matrix product and row operations. Computing the inverse matrix using row operations.
- Permutations. Odd and even permutations. Determinants. Row and column operations on determinants. det(AT)=det(A).
- Minors. Cofactors. Adjoint / adjugate matrix. Computing the inverse matrix using determinants. Cramer's rule for systems with the same number of equations and unknowns.
- Fredholm's alternative. An application: the discrete Dirichlet's problem.
- Coordinate vector space. Linear independence and span.
- Fields: rationals, reals, and complex. Abstract vector spaces. Linear independence and span in abstract vector spaces. Bases and dimensions. Subspaces.
- Linear operators. Matrix of a linear operator relative to given bases. Change of basis. Transition matrices. Similar matrices define the same linear operator in different bases. Example: a closed formula for Fibonacci numbers.
Course materials:
| Lecture | Topics covered | Lecture notes/slides | Homeworks/Tutorials/Solutions |
|---|---|---|---|
| 1 (24/09) | Introduction. Vectors in 2D and 3D, addition, re-scaling, scalar product (dot product). | L1 [PDF] | |
| 2 (25/09) | Vector product (cross product). Multilinearity. Areas and volumes. | L2 [PDF] | |
| 3 (25/09) | Applications of scalar and vector products. Quaternions. Lines and planes in 2D and 3D. | L3 [PDF] | |
| 4 (01/10) | Systems of linear equations. Gauss-Jordan elimination. | L4 [PDF] | |
| 5 (02/10) | Tutorial class on solving systems of linear equations. | ||
| 6 (02/10) | No class at 2pm on October 2. | ||
| 7 (08/10) | Matrix operations. Three definitions of matrix product. | L5 [PDF] | |
| 8 (09/10) | Properties of the matrix product. Elementary matrices. Invertible matrices. | L6 [PDF] | |
| 9 (09/10) | Only a square matrix can be invertible. Computing inverses using elementary row operations. | L7 [PDF] | |
| 10 (15/10) | Permutations. Odd and even permutations. Determinants. Elementary row operations on determinants. | L8 [PDF] | |
| 11 (16/10) | Tutorial class on permutations and determinants | ||
| 12 (16/10) | Determinants and invertibility. Determinant of the product of matrices. | L9 [PDF] | |
| 13 (22/10) | Minors and cofactors. Row expansion of determinants. Adjoint / adjugate matrix. A closed formula for the inverse matrix. | L10 [PDF] | |
| 14 (23/10) | Row expansions, 3D volumes, and cross products. Cramer's rule for solving linear systems. Summary of results on systems with the same number of equations and unknowns. | L11 [PDF] | |
| 15 (23/10) | Finite dimensional Fredholm's alternative. An application: the discrete Dirichlet problem. Vandermonde determinant. | L12 [PDF] | |
| 16 (29/10) | Coordinate vector spaces. Linear independence, span, basis. Linear maps. | L13 [PDF] | |
| 17 (30/10) | Tutorial class on coordinate vector spaces. | ||
| 18 (30/10) | No class at 2pm on October 30. | ||
| Reading week, no classes | |||
| 19 (12/11) | Subspaces of Rn. Two main examples: solution sets to systems of linear equations and linear spans. Relating these two examples. | L14 [PDF] | |
| 20 (13/11) | Abstract vector spaces. Examples. | L15 [PDF] | |
| 21 (13/11) | Simple consequences of properties of vector spaces. Fields: rational, real, and complex. "Coin weighing problem" as an example of field change. | L16 [PDF] | |
| 22 (19/11) | Linear independence and span in abstract vector spaces. Bases and dimensions. Coordinates. | L17 [PDF] | |
| 23 (20/11) | Tutorial class on abstract vector spaces | ||
| 24 (20/11) | Change of coordinates. Transition matrix of coordinate change. | L18 [PDF] | |
| 25 (26/11) | Linear maps. Matrix of a linear map relative to given bases. Composition of linear maps corresponds to the matrix product. | L19 [PDF] | |
| 26 (27/11) | Change of the matrix of a linear map under coordinate changes. Linear transformations. Examples of linear maps. | L20 [PDF] | |
| 27 (27/11) | Examples of linear maps, matrices, and change of bases. | L21 [PDF] | |
| 28 (03/12) | No class on December 3. Homework 9 is due immediately after the tutorial class at 11am on December 4. | ||
| 29 (04/12) | Tutorial class | ||
| 30 (04/12) | Computing Fibonacci numbers. Eigenvectors and eigenvalues. | L22 [PDF] | |
| 31 (10/12) | Eigenvectors and eigenvalues. Example: "porridge problem". | L23 [PDF] | |
| 32 (11/12) | An application of linear algebra: Google PageRank algorithm. | L24 [PDF] | |
| 33 (11/12) | Revision of the module |
MA1212
- Kernels and images, rank and nullity, dimension formula.
- Characteristic polynomials. Eigenvalues and eigenvectors. Diagonalisation in the case when all eigenvalues are distinct.
- Cayley--Hamilton theorem. Minimal polynomial of a linear operator. Examples (operators with A2=A).
- Invariant subspaces. An application: two commuting linear operators have a common eigenvector. Direct sums.
- Normal form of a nilpotent operator. Jordan normal form (Jordan Decomposition Theorem). Applications: closed expressions for Fibonacci numbers and other recursively defined sequences.
- Orthonormal bases; Gram--Schmidt orthogonalisation. Orthogonal complements and orthogonal direct sums. Bessel's inequality.
- Bilinear and quadratic forms. Sylvester's criterion. The law of inertia. Spectral Theorem for symmetric operators.
Course materials
| Lecture | Topics covered | Lecture notes/slides | Homeworks/Tutorials/Solutions |
|---|---|---|---|
| 1 (12/01) | Introduction. Recollections from module 1111: vector spaces, linear maps, linear transformations, change of coordinates, transition matrices. Subspaces. Kernels and images of linear maps. | L1 [PDF] | |
| 2 (14/01) | Rank and nullity of a linear map. Methods of computing rank: row echelon forms, rank-nullity theorem. | L2 [PDF] | |
| 3 (15/01) | Eigenvalues and eigenvectors. Diagonalisation. | L3 [PDF] | HW1 [PDF] |
| 4 (19/01) | Sums and direct sums. Dimensions of sums and intersections. | L4 [PDF] | |
| 5 (21/01) | Example of computing an intersection. Invariant subspaces. Two commuting operators have a common eigenvector. | L5 [PDF] | HW2 [PDF] HW1 solutions [PDF] |
| 6 (22/01) | Tutorial class on intersections and invariant subspaces. | T1 [PDF], T1 solutions [PDF] | |
| 7 (26/01) | Euclidean vector spaces. Examples. Orthonormal bases, Gram-Schmidt orthogonalization process. | L6 [PDF] | HW3 [PDF] |
| 8 (28/01) | No lecture on January 28. Home assignments are still due, submit into the designated cardboard box in the main office of School of Maths before 10am on 28/01 | HW2 solutions [PDF] | |
| 9 (29/01) | Tutorial class on Gram-Schmidt orthogonalization. | T2 [PDF], T2 solutions [PDF] | |
| 10 (02/02) | Lengths and angles in Euclidean spaces. Orthogonal complements. Bessel's inequality. | L7 [PDF] | |
| 11 (04/02) | An application of Bessel's inequality to estimating the sum of inverse squares. Extremal properties of eigenvalues of symmetric matrices. | L8 [PDF] | HW4 [PDF] HW3 solutions [PDF] |
| 12 (05/02) | A real symmetric matrix admits an orthonormal basis of eigenvectors. Orthogonal matrices. Orthogonal matrices in 2D. | L9 [PDF] | |
| 13 (09/02) | Orthogonal matrices in 3D. Quadratic forms (and a motivation from analysis). Bilinear forms. | L10 [PDF] | |
| 14 (11/02) | Symmetric bilinear forms. Canonical form of a symmetric bilinear form. Law of inertia. Theorems of Jacobi and Sylvester. | L11 [PDF] | HW5 [PDF] HW4 solutions [PDF] |
| 15 (12/02) | Tutorial on bilinear and quadratic forms. | T3 [PDF], T3 solutions [PDF] | |
| 16 (16/02) | Proof of the canonical form theorem and the law of inertia. | L12 [PDF] | |
| 17 (18/02) | Proof of the eigenvalues theorem, and the Jacobi theorem. | L13 [PDF] | HW6 [PDF] HW5 solutions [PDF] |
| 18 (19/02) | Proof of the Sylvester's criterion. Hermitian vector spaces. | L14 [PDF] | |
| Reading week, no classes | |||
| 19 (02/03) | Examples of solutions for past tutorials / homeworks. | L15 [PDF] | |
| 20 (04/03) | Midterm exam. For logistics reasons, the class will be separated in two groups. Maths, TSM, and exchange students are to be present at the usual location (Schrödinger lecture theatre) by 8:55am, TP students are to be present at the Synge lecture theatre (Hamilton building) by 8:55am. | Midterm problems [PDF] | Midterm solutions [PDF] |
| 21 (05/03) | Adjoint of a linear transformation. Symmetric (self-adjoint) and unitary transformations. Normal transformations and their diagonalisation. | L16 [PDF] | HW6 solutions [PDF] |
| 22 (09/03) | Normal forms for linear transformations that satisfy φ2=φ or φ2=0. | L17 [PDF] | |
| 23 (11/03) | Examples of relative bases and a normal form for φ2=0. | L18 [PDF] | HW7 [PDF] |
| 24 (12/03) | Normal forms for nilpotent linear transformations (satisfying φk=0). Examples. | L19 [PDF] | |
| 25 (16/03) | Another example of a nilpotent linear transformation. Handling arbitrary linear transformations: kernels of powers and a direct sum decomposition. | L20 [PDF] | |
| 26 (18/03) | Handling arbitrary linear transformations: separating eigenvalues. Jordan decomposition. | L21 [PDF] | HW8 [PDF] HW7 solutions [PDF] |
| 27 (19/03) | Tutorial class on Jordan canonical forms. | T4 [PDF], T4 solutions [PDF] | |
| 28 (23/03) | Uniqueness of Jordan forms. Powers of a Jordan block and applications of Jordan forms to computing powers of matrices. Statement of Cayley-Hamilton theorem. | L22 [PDF] | |
| 29 (25/03) | Proof of Cayley-Hamilton theorem. Examples of dealing with linear recurrences. | L23 [PDF] | HW9 [PDF] HW8 solutions [PDF] |
| 30 (26/03) | Tutorial class on applications of Jordan decomposition. | T5 [PDF], T5 solutions [PDF] | |
| 31 (30/03) | Application of linear algebra to face recognition in images. |
Online note by J. Kun [HTML] Wolfram demonstration of face recognition [HTML] |
|
| 32 (01/04) | Application of linear algebra to JPEG image compression. |
Online notes by J.Rebaza on image compression [PDF] HW9 solutions [PDF] |
|
| 33 (02/04) | Revision of the module. |