MATH 2R03 -- Linear Algebra II

Summer Semester 2018

Main Topics (subject to change)

The relevant sections of the textbook (A&R) are listed in parentheses. Suggested problems from the textbook will also appear here.

A note on suggested textbook problems: as the WileyPLUS assignments are more computational in nature, it is strongly recommended that, in order to practice for the midterms and final examination, you do ALL the "Working with Proofs" (marked as WP below) and "True-False" problems (marked as TF below) from each of the relevant sections of the textbook.

Review of complex numbers.
A&R sections: Since this topic is not covered in sufficient detail in the edition of the textbook that we are using, here is a pdf file of the relevant chapter of the textbook from a former edition, 10.1-10.3.
Suggested problems: 10.1 -- 3, 4e, 4f, 5, 8, 10, 15, 16, 18, 21, 24, 25, 28, 30; 10.2 -- 4, 5, 6, 7, 11, 14, 18, 21, 23, 26c, 40; 10.3 -- 4a, 5, 7, 11, 15 (and the two exercises at the end of the notes from Lecture 1).
Review of vector spaces.
A&R sections: 4.1-4.5.
Suggested problems: 4.1 -- 3-12, WP (especially 26, 27, 28), TF; 4.2 -- 7-13, 16, 17, WP, TF (especially c, f, h, i, j, k); 4.3 -- 3-7, 16, WP (especially 24-29), TF; 4.4 -- 5, 9, 19, 14, 20, 22, 25, 26, WP, TF; 4.5 -- 9, 11, 15, 17, WP (especially 22, 25), TF.
Inner product spaces (real and complex dot products, inner products and inner product spaces (real and complex), norm and distance in inner product spaces, Cauchy-Schwarz inequality, angles and orthogonality in inner product spaces, orthogonal complements).
A&R sections: 6.1-6.2.
Suggested problems: 6.1 -- 1, 4, 10, 11, 16, 22, 38, WP (especially 43, 44, 45), TF; 6.2 -- 3, 7, 9, 11, 13, 17, 18, 22, 23, 26, 33, 35, 37, 39, WP (especially 41-49), TF.

The material from the start of the course up until this point (i.e. the topics listed above here) will form the syllabus for the first midterm. (In particular note that the Review of eigenvalues, eigenvectors and diagonalization, which previously appeared above here on the list of topics, is not included, as we have not done that yet in class.)
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Orthogonal/orthonormal sets and bases, Projection Theorem for inner product spaces, Gram-Schmidt Process for inner product spaces.
A&R section: 6.3, pp. 364-373.
Suggested problems: 6.3 -- 3, 4, 7, 10, 14, 21, 23, 25, 31, 33, 41, WP (especially 51), TF ((a)-(e)).
QR-Decomposition, best approximation, least squares.
A&R sections: 6.3, pp. 374-375, 6.4.
Suggested problems: 6.3 -- 45, 49, TF (f); 6.4 -- 3, 7, 17, 24, WP, TF.
Fourier Series.
A&R section: 6.6.
Suggested problems: 6.6 -- 2, 3, 8, 9, 10, TF.
General Linear Transformations (linear transformations between vector spaces, kernel, range, rank, nullity, compositions, inverses, isomorphisms).
A&R sections: 1.8, 8.1-8.3.
Suggested problems: 1.8 -- 11, 13, 17, 25, 27, 29, WP, TF; 8.1 -- 1, 3, 7, 10, 11, 13, 19, 21, 23, 26, 27, 28, 31, WP, TF; 8.2 -- 1, 3, 5, 7, 15, 17, 19, 21, 23, 29, 31, WP (especially 33, 35, 36), TF; 8.3 -- 3, 5, 6, 15, 21, WP, TF.

The material from the demarcation point above for the first midterm up until this point (i.e. the topics listed between the two sets of dashed lines) will form the main focus of the second midterm. (That is to say, the test will focus on these topics, but you may need to remember some things from your earlier studies!)
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Matrices associated with linear transformations, Change of basis and Similarity (matrix representations of linear transformations including composition and inverse transformations, transition matrices for change of basis, and eigenvalues/eigenvectors, determinant and characteristic polynomial for linear operators).
A&R sections: 8.4-8.5.
Suggested problems: 8.4 -- 1, 3, 5, 13, 19, 21, WP, TF; 8.5 -- 5, 9, 15, 17, 21, WP (especially 29), TF.
Review of eigenvalues, eigenvectors and diagonalization.
A&R sections: 5.1-5.3.
Suggested problems: 5.1 -- 7, 11, 12, 13, 14, 24, 25, WP (especially 32, 33, 34, 35, 36), TF; 5.2 -- 5, 7, 8, 17, 27, WP (especially 39, 40, 41), TF; 5.3 -- 12, 14-18, WP (especially 33, 36), TF.
Orthogonal matrices
A&R section: 7.1
Suggested problems: 7.1 -- 5, 7, 11, 12, 22, WP (especially 25, 26, 27, 30), TF.
Orthogonal and unitary diagonalization, unitary matrices, real symmetric, Hermitian and normal matrices.
A&R sections: 7.2, pp. 409-412, 7.5 (but not including either skew-symmetric or skew-Hermitian).
Suggested problems: 7.2 -- 7, 9, 11, 21, 22, WP (especially 25, 26, 29, 30, 31), TF; 7.5 -- 1, 7, 11, 13, 17, 25, 27, 33, 34, WP (especially 39, 41), TF (especially (a)-(d)).
Quadratic forms, conic sections, optimization of quadratic forms, definiteness of quadratic forms, Hessian form of the second derivative test.
A&R sections: 7.3-7.4.
Suggested problems: 7.3 -- 1, 7, 13, 15, 17, 21, 27; WP, TF; 7.4 -- 3, 5, 7, 13, 15, 18, 20, 21, WP, TF.

Schedule of Lectures

Summaries of lectures and notes from class will be posted here as the semester progresses. The lecture notes below are typically augmented versions of what was presented on screen during class; the augmentations are principally to be found in the form of extra comments (and corrections) written in orange. You are strongly advised to go through these notes alongside your own notes carefully to make sure that you understand everything that was discussed!

Lecture 1 (19 June 2018) -- Review of Complex Numbers. In particular: definitions, addition, multiplication, division, complex conjugates, polar form, exponentiation, roots. Regarding the exercises set right at the end of class: the answers are now available in this file . ~~answers will be posted before the start of Lecture 3.~~ Do have a go at them before reading the solutions!
Lecture 2 (21 June 2018) -- Review of Vector Spaces. In particular: axioms for real or complex vector spaces, examples, subspaces, linear independence, spanning, bases, dimension, coordinates, dot product.
Lecture 3 (26 June 2018) -- Review of Dot Products; and Inner Product Spaces. In particular: properties of the real and complex dot products, definition of inner product and inner product space (real and complex), examples of inner product spaces, definitions of norm and distance in inner product spaces.
Lecture 4 (28 June 2018) -- More properties of inner products, norms and distances, Cauchy-Schwarz inequality, angles and orthogonality in inner product spaces, including orthogonal complements.
Lecture 5 (3 July 2018) -- Properties of the orthogonal complement, orthogonal/orthonormal sets and bases, Projection Theorem for inner product spaces, Gram-Schmidt Process for inner product spaces.
Lecture 6 (5 July 2018) -- Midterm 1 (see Assessment Methods). QR Decomposition, projection as best approximation, least squares solutions. Regarding the QR Decomposition exercise set in the middle of class: the answer is now available in this file .
Lecture 7 (10 July 2018) -- Fourier Series. Properties of matrix transformations. Definition and examples of Linear Transformations (and Operators), Kernel and Range of a linear transformation.
Lecture 8 (12 July 2018) -- Rank--Nullity Theorem, inverses and compositions of linear transformations, isomorphism between vector spaces, introduction to matrices associated with linear transformations.
Lecture 9 (17 July 2018) -- matrix representations of linear transformations including composition and inverse transformations, transition matrices for change of basis, (very brief) Review of eigenvalues, eigenvectors and diagonalization, and eigenvalues/eigenvectors, determinant and characteristic polynomial for linear operators.
Lecture 10 (19 July 2018) -- Midterm 2 (see Assessment Methods). Orthogonal matrices, properties of orthogonal matrices and of orthonormal bases.
Lecture 11 (24 July 2018) -- Orthogonal diagonalization of real symmetric matrices, properties of unitary matrices, unitary diagonalization, Hermitian matrices and normal matrices.
Lecture 12 (26 July 2018) -- Quadratic forms, conic sections, optimization of quadratic forms, definiteness of quadratic forms, Hessian form of the second derivative test.
Lecture 13 (31 July 2018) -- Review only. Topics covered: Linear Transformations (Rank--Nullity Theorem, onto, 1-1 transformations); some proofs involving linear transformations; how to find matrices representing linear transformations with respect to certain bases; Unitary(/Orthogonal) diagonalization; Projection as Best Approximation.
Lecture 14 (2 August 2018) -- Final Examination (see Assessment Methods).

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