Course webpage for Math 121

Please let me know if any of the links fail to work, or if you spot any errors, or if anything is unclear.

Final

The final exam will be on Thursday 20th May at 2.15pm in Harvard 103. You are responsible for all material covered in class, except for any material which was specifically described as non-examinable. In addition, you are responsible for any reading that you needed to do to do any of your homework, plus the following material from the textbook:

Sections 1.1-1.6 (vector spaces, subspaces, linear combinations, linear independence, bases, dimension). You do not need to know how to prove that every infinite dimensional vector space has a basis.
Sections 2.1-2.7 (linear transformations, null space, range, matrices, invertibility and isomorphisms, change-of-basis, dual spaces, applications to differential equations). You do not need to know about the duals of infinite-dimensional vector spaces.
Material from Chapter 3 as described in this email.
Section 4.4 (but you don't need to know how to prove any of these properties of determinants).
Sections 5.1, 5.2 and 5.4 (eigenvalues, eigenvectors, diagonalizability, invariant subspaces, the Cayley-Hamilton theorem).
Sections 7.1 and 7.2 (Jordan canonical form). You do not need to know anything about "dot diagrams".
Sections 6.1 to 6.4 omitting pages 370-372, section 6.5 as far as the end of p383, and section 6.7 omitting the material on Polar Decomposition (inner product spaces, Gram-Schmidt, orthogonal complements, adjoints, self-adjoint operators and their diagonalizability, Singular Value Decomposition and pseudoinverse).

You should expect roughly half the exam to cover material that was assessed on the midterms, and half the exam to concentrate on the material that we covered since the second midterm. A sample final is available here. The solution is here.

The sample second midterm is available here. Solutions are here:

The sample first midterm is available here. Solutions are here:

Homework

Homework assignments and homework solutions will appear here.

Assignments with a "Due date" of "n/a" are optional; if you turn them in then they will be graded but they will not affect your homework score. If you want these extra problems to be graded then you must turn them in at the same time as the corresponding homework set.

Assignment date	Due date	Problems
Thursday 5th February	Tuesday 10th February	Prerequisites refresher (see handouts below)
	Thursday 12th February	1.2 Q12,13,16,18; 1.3 Q10,12,23,26; 1.4 Q10,15
	n/a	1.2 Q1,22; 1.3 Q1; 1.4 Q3,7,12
	n/a	1.2 Q20; 1.3 Q19,28; 1.4 Q1(c)
Thursday 12th February	Thursday 19th February	1.5 Q9,13,18; 1.6 Q2(a-b),14,15,17,21,29; 1.7 Q3
	n/a	1.5 Q1,5; 1.6 Q1,22; 1.7 Q1
	n/a	1.5 Q8,20; 1.6 Q26,34; 1.7 Q2
Thursday 19th February	Thursday 26th February	1.6 Q31,32,33,34; 2.1 Q9,13,19,35; 2.2 Q3, 13
	n/a	2.1 Q1,6,11; 2.2 Q1,8
	n/a	2.1 Q14,15; 2.2 Q5,9,11
Thursday 26th February	Thursday 4th March	2.1 Q28, 31; 2.2 Q9,11,14,16; 2.3 Q9,11,12,16

New instructions for homework: Assignments with a due date of "extra" are optional. You are under no obligation to do any of them, and you will not be penalized for not doing so. However, you may replace any homework question which appears in bold with any of the questions with due date "extra".

Assignment date	Due date	Problems	Solutions
Thursday 4th March	Saturday 13th March	2.4 Q14,16,17; 2.5 Q6(a,c),7,8; 2.6 Q4,6,8,10	rest:
Thursday 4th March	extra	2.4 Q23,25; 2.5 Q14
Thursday 11th March	Thursday 18th March	2.6 Q3(a),7,13,15,20 (finite-dim. case only); 1.3 Q31(a-c,d)
Thursday 11th March	extra	2.6 Q19, 20 (infinite-dim. case)
Thursday 18th March	Thursday 25th March	2.1 Q40; 2.4 Q24; 2.7 Q2,12,13,18; 3.2 Q6(d-e),17; 3.3 Q3(d,g),10
Thursday 18th March	extra	2.7 Q9; 3.2 Q14
Friday 26th March	Thursday 8th April	5.1 Q1,3(c,d),4(f,g,h),8,14,17; 5.2 Q1,8,12,13
Friday 26th March	extra	5.1 Q22,23
Saturday 10th April	Friday 16th April	please see special instructions below
	regular	5.2 Q2(d,f),7; 5.4 Q13,15,17,21,27,28,29,30
	Markov chains	5.3 Q6,7,8(a,d),11,12,20,21,22
Saturday 17th April	Friday 23rd April	7.2 Q3(omit part b),4(b,c),5(a,d),6,8,10,12,17
Saturday 24th April	Friday 30th April	6.1 Q8,11,12; 6.2 Q6,10,11,13,14,16,18
Saturday 24th April	extra	6.2 Q15,22
Tuesday 4th May	Tuesday 11th May	6.3 Q3,6,11,15; 6.4 Q14; 6.7 Q3(a,b),9,17,21,22
Tuesday 4th May	extra	6.3 Q21,23

Special instructions for homework due Friday 16th April: Several people expressed interest in the material on Markov chains, but there will not be time to cover it in class. If you want to, you may replace any of the questions on the homework (which are marked "regular") with any of the questions marked "Markov chains". If you do this, you will need to read section 5.3 in the textbook. Full disclosure: there will be no questions on further homework or on the final about the material on Markov chains (but it is, of course, an interesting and useful application of the material on diagonalization).

Course outline

Overview: syllabus, motivation; Vector spaces: axioms, examples. Vector spaces: examples, direct sums; Fields; Subspaces; Linear combinations and span. Linear independence; Bases: definition, conditions for being a basis. Steinitz exchange lemma; Dimension and its consequences; Bases for infinite-dimensional vector spaces. Linear transformations: definition, examples and simple properties; Kernel and image; Dimension theorem. Sets of linear transformations as vector spaces; Representation of linear transformations by matrices. Invertibility and isomorphisms; Identifying an n-dimensional vector space with F^n via a basis. Change of basis. Dual spaces: definition, dual basis, the transpose of a linear transformation. Catch-up; Applications to differential equations. Constructions: quotient spaces, tensor product; Applications. Elementary row and column operations; Applications: rank, matrix inversion, solving linear systems. Determinants (via volumes). Eigenvalues and eigenvectors; Characteristic polynomial; Diagonalization. Factoring polynomials over R and C; Algebraic and geometric multiplicity; Conditions for diagonalizability. Spring BreakSpring Break Jordan canonical form (statement only); Applications: differential equations, resonance. Invariant subspaces; Cyclic subspaces; Cayley-Hamilton theorem. Jordan canonical form: existence. Jordan canonical form: existence (continued), uniqueness; Minimal polynomial. Inner product spaces and norms; Examples: finite-dimensional and infinite-dimensional. Gram-Schmidt process; QR factorization; Matrix entries; Orthogonal projections. Riesz representability theorem; Adjoints and their matrices; Application: least squares curve fitting. Self-adjoint operators: definition and properties, characterization. Unitary and orthogonal operators; the Spectral Theorem; Singular value decomposition. Applications: polar decomposition, pseudoinverse; Bilinear forms; Application: the Second Derivative Test.

Date	Reading	Topics
Thursday 5th February	§1.2
Tuesday 10th February	§§1.3-1.4, Appendix C
Thursday 12th February	§§1.5-1.6
Tuesday 17th February	§§1.6-1.7
Thursday 19th February	§2.1
Tuesday 24th February	§§2.2-2.3
Thursday 26th February	§§2.3-2.4
Tuesday 2nd March	§2.5
Thursday 4th March	§2.6
Tuesday 9th March	n/a
Thursday 11th March	n/a
Tuesday 16th March	§§3.1-3.2
Thursday 18th March	Chapter 4
Tuesday 23rd March	§§5.1-5.2
Thursday 25th March	§5.2
Tuesday 30th March	n/a
Thursday 1st April	n/a
Tuesday 6th April	§5.3 (optional)
Thursday 8th April	§5.4
Tuesday 13th April	§7.1
Thursday 15th April	§§7.1-7.3
Tuesday 20th April	§6.1
Thursday 22nd April	§6.2
Tuesday 27th April	§6.3
Thursday 29th April	§6.4
Tuesday 4th May	§6.6
Thursday 6th May	§6.7

Handouts

Handouts will appear here.

Thursday 5th February		Policies and practicalities
		First homework
		Suggested reading
		Survey (please fill out and turn in if you have not already done so)
Tuesday 10th February		Axioms summary
		Suggested reading: complex numbers
		Suggested reading: fields (coming soon!)
Thursday 12th February		Policies and practicalities (updated)
Thursday 8th April		Lecture notes
Tuesday 20th April		End of the proof of the Lemma