Grading

Final Exam (Tuesday, May 12; 11AM--2PM; SEC 203) final exam & solutions

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Week 14 (4/27--5/1)

Midterm Exam 2 (April 30) midterm 2 & solutions

Review and the Q&A Session (April 28)

Week 13 (4/20--4/24)

Quiz 5 (April 23) quiz 5 & solutions

Review (April 21 & 23) notes

Lecture 22 (April 21) 5.3 Matrix Limites and Markov Chains notes

Key concepts: Matrix Limits, Existence of Limits

Reading: Section 5.3.

Week 12 (4/13--4/17)

Assignment 12 (April 21): 5.2(3, 6, 7, 12, 13, 18)

Lecture 21 (April 16) 5.2 Diagonalizability. notes

Key concepts: algorithm to check diagonalizability; split or non-split polynomials; multiplicity of roots of characteristic polynomial; dimension of eigenspace.

Reading: Section 5.2.

Lecture 20 (April 14) 5.1 Eigenvalues and eigenvectors (cont.). notes

Key concepts: eigenvectors and eigenvalues; diagonalizability; characteristic polynomial; eigenvalues equal roots of characteristic polynomial.

Reading: Section 5.1.

Week 11 (4/6--4/10)

Quiz 4 (April 9) quiz 4 & solutions

Assignment 11 (April 14): 5.1(5, 8, 9, 12, 14, 20)

Some remarks on selected problems

Lecture 18 & 19 (April 7 & 9) 5.1 Eigenvalues and eigenvectors. notes

Key concepts: eigenvectors and eigenvalues; diagonalizability; characteristic polynomial; eigenvalues equal roots of characteristic polynomial.

Reading: Section 5.1.

Review (April 7) notes

Week 10 (3/30--4/3)

Assignment 10 (April 7): 4.2(1, 3, 5, 9, 14, 23); 4.3(4, 9, 11, 13, 15, 21)

Some remarks on selected problems

Lecture 17 (April 2) 4.3 Properties of Determinants notes

Key concepts: determinant of product of matrices is product of determinants; determinant is nonzero if and only if matrix is invertible; determinants of elementary matrices; determinant of transpose; effect of column operations.

Reading: Section 4.3.

Lecture 16 (March 31) 4.2 Determinants of Order n. notes

Key concepts: geometry of determinants of nxn-matrices; inductive definition of determinants in general; linearity with respect to fixed row; effect of row operations.

Reading: Section 4.2.

Quiz 3 (March 31, New Date) quiz 3 & solutions

Week 9 (3/23--3/27)

Assignment 9 (March 31): 3.4(1, 2, 4, 7, 10, 12); 4.1(2, 3, 4, 7, 9, 12)

Review (March 26) notes

Lecture 15 (March 26) 4.1 Determinants of Order 2. notes

Key concepts: geometry of determinants of 2x2-matrices; linearity with respect to fixed row.

Reading: Section 4.1

Lecture 14 (March 24) 3.4 Solving systems of linear equations. Computational. notes

Key concepts: Gaussian elimination and reduced row echelon form; solving systems of linear equations in reduced row echelon form.

Reading: Section 3.4.

Week 8 (3/9--3/13)

Assignment 8 (March 24): 3.2(3, 5, 6, 7, 14, 17); 3.3(2, 3, 8, 10, 12, 14)

Lecture 13 (March 12) 3.3 Solving systems of linear equations. Theoretical. notes

Key concepts: consistent and inconsistent systems; homogeneous and non-homogeneous systems; relation of solutions and null space; criteria for existence of solutions.

Reading: Section 3.3.

Lecture 12 (March 10) 3.2 Rank and Inverses notes

Key concepts: rank of matrix; simplifying matrices; calculating inverses.

Reading: Section 3.2.

Week 7 (3/2--3/6)

Assignment 7 (March 10): 3.1(2, 4, 5, 7, 10, 12)

Midterm Exam 1 (March 5, New Date) midterm 1 & solutions

Quiz 2 (March 3, New Date) quiz 2 & solutions

Lecture 11 (March 3) 3.1 Elementary matrix operations. notes

Key concepts: elementary matrices and operations.

Reading: Section 3.1.

Week 6 (2/23--2/27)

Assignment 6 (March 3): 2.5(2, 4, 7, 9, 12, 14)

Some remarks on selected problems

Review (February 26) notes

Lecture 10 (February 24) 2.5 Change of bases, 2.6 Dual spaces notes

Key concepts: change of coordinate matrices; similar matrices; dual spaces and dual bases; transposes

Reading: Section 2.5 and 2.6.

Week 5 (2/16--2/20)

Assignment 5 (February 26): 2.3(2, 10, 12, 13, 16, 23); 2.4(2, 3, 6, 7, 15, 17)

Some remarks on selected problems

Lecture 9 (February 19) 2.4 Invertibility and Isomorphisms notes

Key concepts: isomorphisms and inverses; every finite dimensional vector space is isomorphic to coordinate space.

Reading: Section 2.4.

Lecture 8 (February 17) 2.3 Composition of linear transformations notes

Key concepts: compositions of maps; basic properties of compositions; multiplication of matrices.

Reading: Section 2.3.

Week 4 (2/9--2/13)

Assignment 4 (February 19): 2.1(6, 11, 12, 20, 24, 35); 2.2(2, 6, 8, 10, 12, 16)

Some remarks on selected problems

Lecture 7 (February 12) 2.2 Properties of linear transformations, Matrices notes

Key concepts: injective, surjective, bijective maps of sets; isomorphisms of vector spaces; coordinates with respect to a basis; matrices with respect to bases; the vector space of linear transformations.

Reading: Section 2.2.

Lecture 6 (February 10) 2.1 Linear transformations notes

Key concepts: maps of sets; linear maps of vector spaces; kernels and images; nullity and rank, Dimension Theorem.

Reading: Section 2.1.

Quiz 1 (February 10, New Date) quiz 1 & solutions

Week 3 (2/2--2/6)

Assignment 3 (February 10): 1.6(4, 9, 17, 20, 26, 34)

Some remarks on selected problems

Review (February 5) notes

Lecture 5 (February 3) 1.6 bases and dimension notes

Key concepts: finite bases; constructing generating sets; finding bases; Replacement Theorem; dimension.

Reading: Section 1.6.

Week 2 (1/26--1/30)

Assignment 2 (February 3): 1.4(3, 5, 10, 12, 15, 17); 1.5(3, 9, 11, 12, 13, 17)

Some remarks on selected problems

Lecture 4 (January 29) 1.5 Linear dependence and independence notes

Key concepts: linear dependence and independence; Properties of Linear Dependence and Linear Independence

Reading: Section 1.5.

Lecture 3 (January 27) 1.4 Linear combinations and systems of linear equations notes

Key concepts: linear combinations; solving systems by row reductions; span of a subset; generating sets

Reading: Section 1.4

Week 1 (1/20--1/23)

Assignment 1 (January 29): 1.2(6, 8, 13, 16, 17, 21); 1.3(8, 9, 19, 23, 29, 30)

Some remarks on selected problems

Lecture 2 (January 22) 1.3 Subspaces notes

Key concepts: subspace characterizations; intersection of subspaces; sum of subspaces; direct sum of vector spaces; transpose of matrix, symmetric and skew- or anti-symmetric matrices; trace of matrices, diagonal matrices; upper/lower triangular matrices.

Reading: Section 1.3.

Lecture 1 (January 20) 1.2 Vector Spaces notes

Key concepts: vector space axioms; row vectors, column vectors, matrices; functions, polynomials.

Reading: Sections 1.1 & 1.2, Appendix C.