Week 1: Vectors, vector spaces, span, linear independence,
bases (corrected, Aug 19 2008; thanks to Radhakrishna Bettadapura for the correction) [Further correction, Jul 16 2014: On page 25, last sentence of third paragraph, "course" should be "of course", and on page 35, "which both spans S" should be "which both spans V". Thanks to Luqing Ye for the correction.] [Further correction, Aug 8 2014: On Page 17, first bullet point, "every vector in V is also a vector in W" should be "every vector in W is also a vector in V", and a right parenthesis is missing in the first paragraph. Thanks to Cristhian Gonzales for the correction.] [Further correction, Sep 10 2015: in the third bullet point on page 3, all occurrences of 1/m should be m.]
Week 4: Matrix multiplication, Invertibility, Isomorphisms
(corrected, Oct 28) [Further errata (Nov 21, 2016): In the last line of page 15, "invertible" should be "injective". Thanks to Sasha Illarionov for the correction. (May 23, 2019) On page 8, "vectors v in V" should be "vectors v in X". Thanks to Rohan Varma for the correction. (Nov 14, 2019) In the display in the last line of page 18 and the first lines of page 19, all the indices in B should be swapped, e.g. B_{1n} should be B_{n1}, B_{m1} should be B_{1m}, and so forth. (Thanks to Shirin Ebrahimzadeh for the correction). (Apr 10, 2021) In page 7, c_m2 should be c_n2 and b_m2 should be b_n2. (Thanks to Nianchen Liu for the correction).]
Week 5: Co-ordinate change, midterm review [Correction, May 2016: On page 2, in the first example, the column vector (4, -1) should instead be (4,1). Thanks to Wilson Sov for the correction. Further correction, Apr 7 2021: In pages 5-6, many of the column vectors need to have their entries transposed. Thanks to Abishek Kundu for the correction.]
Week 6: Review of elementary row operations, rank,
determinants (Corrected, Nov 25. Nov 15, 2019: On page 7, line 7, T should be replaced by QT instead of RT. (Thanks to Shirin Ebrahimzadeh for the correction.) Apr 19 2021: in Page 21, in properties 2 and 3, "the matrix B" should be "the matrix A". (Thanks to Nianchen Liu for the correction.)
Week 7: Eigenvalues, Eigenvectors, diagonalization
(corrected, Nov 25. Further correction, Jul 26 2017: in the fourth display of page 4, the matrix entries reading -c and -b should be swapped. Thanks to Lin Yubin for the correction.)
Week 8: Characteristic polynomials, inner products
(corrected, Dec 1) [Further correction, Nov 17 2010: on the bottom of page 24, the subscript 2 on the norm of w should not be present. Thanks to Michael Smith for the correction. (Jun 26 2021) On the bottom of page 8, T-3I and T-5I should be interchanged, and the second eigenvalue is 3 rather than 5 (thanks to Nianchen Liu for the correction).]
Week 9: Norms, orthogonal bases, orthognalization,
orthogonal complements[Correction: on the end of page 3, all series starting with v_2 should instead start with v_1. (June 26 2021) On page 19, the second "normal, but not self-adjoint" should be "normal and self adjoint". Thanks to Nianchen Liu for the correction.].
Week 10: Adjoints, normal and self-adjoint operators, final review Note that the material on adjoints, normal operators, and self-adjoint operators will not be covered in the final, but this material is still worth reading, especially if you are taking Math 115B or any other course which requires 115A as a prerequisite. [Further correction, Oct 26 2010: On page 2, example 3, "thu" should be "thus", and on page 19, "ahve" should be "have". On page 3, in the statement of the Riesz representation theorem, "w \in W" should be "w \in V"; also, a right bracket is missing in the middle of the proof. Thanks to Michael Smith, Yu Cao, and Shijia Yu for the corrections. Apr 26, 2021: on the top line of page 15, T^* v = TT^* v should be TT^* v = T^* v. (Thanks to Abishek Kundu for the correction.)]
Sample exams and solutions
Sample midterm. and solutions
(Note: this is from an Honors class of 115A; I've toned down the difficulty
a bit, but this is still more challenging than the actual midterm).
Sample
final. Note: "Kernel" is the same thing as "Null space". For the purposes
of this class, the field of scalars F can always taken to be the real numbers
R. Question 6 is a little tricky, and Questions 7,8 should be skipped altogether (they cover
some material not covered in this course, and besides Q7 has some typos). Solutions are available here.
Another practice final. Note that this is from a 115AH course and so covers some material beyond what is covered in this course. In particular, Questions 4,5,6,10 require material not covered in this course,
Question 12 uses material from the Week 10 notes (which are not covered in the final), and Question 8 is doable but may be a little tricky for a non-honors
class. However, the other questions give a reasonable indication of what
to expect on this class's final.