Math122–001 Linear Algebra I, Winter 2012

Lecture Summaries

Summaries of the topics covered in each class will be placed here, after the class. The numbers in parentheses indicate section numbers from the course textbook.

Week 1

Friday 6 January: Introduction to vectors in R², R³ and Rⁿ; geometric interpretation. (1.1)

Week 2

Monday 9 January: Rules for vectors: equality, addition, scalar multiplication, subtraction. The norm of a vector; distance between two points in Rⁿ. (1.1)

Wednesday 11 January: Facts about norms: norms of scalar multiples, normalization and unit vectors. The dot product: definition, finding angles between vectors, properties. (1.1, 1.2)

Friday 13 January: Standard unit vectors i, j, k. The cross product: definition, examples, properties. (1.2)

Week 3

Monday 16 January: Equation of a line in R²; normal vectors; lines in R³; parametric equations. (1.3)

Wednesday 18 January: Parallel, perpendicular and intersecting lines in R³; planes in R³. (1.3)

Friday 20 January: Planes in R³ (continued): determining a plane from three points; intersection of a line and a plane in R³; intersections of planes. (1.3)

Week 4

Monday 23 January: Linear equations; systems of linear equations; solutions to systems of linear equations; geometric interpretation. (2.2)

Wednesday 25 January: Solving systems of linear equations: the coefficient matrix and augmented matrix of a linear system; elementary row operations; row-echelon form and reduced row-echelon form (RREF) of a matrix. (2.3)

Friday 27 January: Using Gauss–Jordan elimination to solve systems of linear equations: examples with unique solution, no solution and infinitely many solutions; free variables. (2.3)

Week 5

Monday 30 January: Further examples of Gauss–Jordan elimination. (2.3) Homogeneous systems of linear equations.

Wednesday 1 February: Matrices: definition; equality, addition, subtraction, scalar multiplication, matrix multiplication. (3.1)

Friday 3 February: Some special matrices: the zero matrix, diagonal matrices, the identity matrix. Laws for matrix operations; matrix multiplication is not commutative. (3.1)

Week 6

Monday 6 February: Row and column matrices; the matrix form of a linear system; the cancellation property does not hold for matrices; the inverse of a matrix. (3.2)

Wednesday 8 February: Formula for the inverse of a 2x2 matrix; elementary matrices and row operations; algorithm for finding the inverse of a matrix. (3.2)

Friday 10 February: Midterm Test 1.

Week 7

Monday 13 February: Examples of finding the inverse of a matrix (3.2); homogeneous linear systems; rank of a matrix; properties of invertible matrices. (3.3)

Wednesday 15 February: Determinants: 3x3 determinants by cofactor expansion; 4x4 determinants. Upper-triangular, lower-triangular and diagonal matrices. (4.1)

Friday 17 February: Determinants of triangular matrices; effect of row operations on determinants; using row operations to find determinants; determinants of invertible matrices. (4.1, 4.2)

Reading Week

No classes this week.

Week 8

Monday 27 February: Class cancelled due to illness.

Wednesday 29 February: Determinants of elementary matrices; determinants of products, sums and scalar multiples of matrices.

Friday 2 March: Linear combinations; linear dependence and independence. (5.1)

Week 9

Monday 5 March: Linear independence (continued); geometric interpretation; subspaces of Rⁿ. (5.1, 5.2)

Wednesday 7 March: Subspaces of Rⁿ: more examples; lines and planes as subspaces of R³. (5.2)

Friday 9 March: Midterm Test 2.

Week 10

Monday 12 March: The span of a set of vectors; spanning sets for Rⁿ. (5.2)

Wednesday 14 March:Review of Test 2. Bases for Rⁿ. (5.3)

Friday 16 March: Bases for subspaces of Rⁿ. (5.3)

Week 11

Monday 19 March: Informal discussion of vector spaces; dimension of a vector space or subspace (5.3/6.2); orthogonal and orthonormal bases.

Wednesday 21 March: Orthogonal projections; the Gram–Schmidt process for finding orthogonal bases. (6.4: note that the treatment is more advanced than we need)

Friday 23 March: The row space, column space and null space of a matrix; rank and nullity; the Rank–Nullity Theorem. (5.3, 5.4)

Week 12

Monday 26 March: Linear transformations: definition and examples; the standard matrix. (7.1)

Wednesday 28 March: Further examples of linear transformations: the zero and identity transformations; scalar multiplication (dilation/contraction); orthogonal projection; rotation and reflection (in R²). Composition of linear transformations; connection with matrix multiplication. (7.1, 7.2)

Friday 30 March: Composition of linear transformations: further examples. One-to-one, onto and invertible linear transformations; connection with invertible matrices. (7.2)

Week 13

Monday 2 April: Reminder: properties of linear transformations. Eigenvalues and eigenvectors of linear transformations and matrices: definition; method for computing eigenvalues; properties of eigenvalues. (4.4, 8.1)

Wednesday 4 April: Eigenvectors and eigenspaces of matrices: definition and examples. (8.1)

Friday 6 April: No class due to Good Friday holiday.

Week 14

Monday 9 April: Review class: subspaces; finding bases for subspaces.

Wednesday 4 April: Review class: linear transformation; standard matrix; one-to-one and onto linear transformations.

Page last updated: 11 April 2012.

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