Math122–003 Linear Algebra I, Fall 2009

Lecture Summaries

The numbers in parentheses indicate section numbers from the course textbook.

Week 1

Wednesday 9th September: Introduction to vectors in R² and R³; geometric interpretation (3.1).

Friday 11th September: Rules for vectors: equality, addition, scalar multiplication, subtraction (3.1). The norm of a vector; distance between two points in Rⁿ (3.2).

Week 2

Monday 14th September: Facts about norms; normalization and unit vectors (3.2). Dot products: definition, finding angles between vectors, properties (3.3).

Wednesday 16th September (shortened class due to fire alarm): Equation of a line in R²; normal vectors; orthogonal projections (3.3).

Friday 18th September: Distance from a point to a line; lines in R³, parametric equations (3.5).

Week 3

Monday 21st September: Intersection of two lines (3.5); standard unit vectors i, j, k; cross products (definition, examples, properties) (3.4).

Wednesday 23rd September: Planes in R³: equations in point-normal form and general form, examples; lines and planes in R³ (3.5).

Friday 25th September: Intersection of lines and planes in R³; distance from a point to a plane in R³; distance from a point to a line in R³; intersection of two and three planes in R³ (all 3.5).

Week 4

Monday 28th September: Linear equations; solutions to linear equations; systems of linear equations; solution sets (all 1.1); geometric interpretation (3.5).

Wednesday 30th September: Finding solutions to systems of linear equations: the augmented matrix of a linear system; elementary row operations (1.1).

Friday 2nd October: Row-echelon form and reduced row-echelon form (RREF) of a matrix; using RREF to solve linear systems; examples with unique solution, infinitely many solutions, no solutions (1.2).

Week 5

Monday 5th October: Gauss–Jordan elimination; homogeneous systems of linear equations (1.2).

Wednesday 7th October: Introduction to vector spaces: definition and examples (5.1).

Friday 9th October: More on vector spaces: closure under addition and scalar multiplication; more examples/non-examples (5.1).

Week 6

Monday 12th October: NO CLASS (due to Thanksgiving).

Wednesday 14th October: Subspaces; the Subspace Test; examples (all 5.2).

Friday 16th October: TEST 1.

Week 7

Monday 19th October: More examples of subspaces; linear combinations; the span of a set of vectors (all 5.2).

Wednesday 21st October: Properties of the span, including that the span is a subspace; spanning sets: definition and examples (all 5.2).

Friday 23rd October: Linearly independent and linearly dependent sets of vectors: definition and examples (5.3).

Week 8

Monday 26th October: More on linear independence: properties of linearly dependent sets (5.3). Bases for vector spaces: definition and examples (5.4).

Wednesday 28th October: Bases for subpaces; the dimension of a vector space: definition and examples (all 5.4).

Friday 30th October: Orthogonal and orthonormal bases; the Gram–Schmidt Process (6.3: note that the treatment given by the textbook is much more advanced that what we did).

Week 9

Monday 2nd November: Orthonormal bases for subspaces; more examples of the Gram–Schmidt Process (6.3).

Wednesday 4th November: Matrices; notation for matrices; operations on matrices: addition, subtraction, scalar multiplication, matrix multiplication (all 1.3).

Friday 6th November: The zero matrix and identity matrix; row and column matrices; row space and column space of a matrix; matrix form of a linear system; null space (1.3, 1.4, 5.5).

Week 10

Monday 9th November: The trace and transpose of a matrix; the cancellation property does not hold for matrices; the inverse of a matrix; formula for 2x2 inverses (1.3, 1.4).

Wednesday 11th November: NO CLASS (due to Remembrance Day).

Friday 13th November: TEST 2.

Week 11

Monday 16th November: Elementary matrices; relationship with row operations and matrix inverses; method for computing the inverse of an n-by-n matrix (all 1.5).

Wednesday 18th November: Examples of computing the inverse (1.5); matrix powers, negative powers, laws of exponents, matrix polynomials (1.4).

Friday 20th November: Review of Test 2; determinants, cofactor expansion (2.1).

Week 12

Monday 23rd November: Cofactor expansion about other rows and columns (2.1); 4x4 determinants by cofactor expansion (2.1); upper-triangular, lower-triangular and diagonal matrices (1.7) and their determinants; finding determinants by row operations (2.2).

Wednesday 25th November: Determinants of: invertible matrices, elementary matrices, products of matrices, inverses, sums and transposes (2.2/2.3).

Friday 27th November: Finding bases for the row space, column space and null space of a matrix (5.5). Linear transformations: definition and examples (4.2/4.3).

Week 13

Monday 30th November: Linear transformations: more examples; standard matrices; the zero and identity transformations; contraction and dilation; orthogonal projections as linear transformations (all 4.2).

Wednesday 2nd December: Rotations and reflections as linear transformations in R² (4.2); composition of linear transformations and matrix multiplication (4.2); one-to-one linear transformations (4.3).

Friday 4th December: Eigenvalues and eigenvectors of linear transformations and matrices; finding eigenvalues by solving the characteristic equation; eigenspaces and finding bases for eigenspaces (all 7.1; also mentioned briefly in 2.3 and 4.3).

Week 14

Monday 7th December: More examples of calculating eigenvalues and bases of eigenspaces; eigenvalues of invertible matrices; the sum and product of eigenvalues; eigenvalues of triangular matrices (all 7.1).

Wednesday 9th December: Review class.

Page last updated: 7th December 2009.

Back to course homepage