MATH 2318 Linear Algebra

Learning Objectives

The objectives of Linear Algebra are:

(1) The student will understand the mathematical concepts and terminology involved in Linear Algebra.

(2) The student will gain an acceptable level of computational proficiency involving the procedures in Linear Algebra.

(3) The student will understand the axiomatic structure of a modern mathematics subject and learn to construct simple proofs.

(4) The student will be able to apply his or her knowledge to applications of Linear Algebra.

The topics that will enable this course to meet its objectives are:

(i) the basic arithmetic operations on vectors and matrices, including inversion and determinants, using technology where appropriate;

(ii) solving systems of linear equations, using technology to facilitate row reduction;

(iii) the basic terminology of linear algebra in Euclidean spaces, including linear independence, spanning, basis, rank, nullity, subspace, and linear transformation;

(iv) the abstract notions of vector space and inner product space;

(v) finding eigenvalues and eigenvectors of a matrix or a linear transformation, and using them to diagonalize a matrix;

(vi) projections and orthogonality among Euclidean vectors, including the Gram-Schmidt orthonormalization process and orthogonal matrices;

(vii) the common applications of Linear Algebra, possibly including Markov chains, areas and volumes, Cramer's rule, the adjoint, and the method of least squares;

(viii) the nature of a modern mathematics course: how abstract definitions are motivated by concrete examples, how results follow from the axiomatic definitions and are specialized back to the concrete examples, and how applications are woven in throughout. This course will present various "characterization" theorems (eg. characterizing isomorphic finite-dimensional vector spaces by their dimension and characterizing invertible matrices by various criteria);

(ix) basic proof and disproof techniques, including mathematical induction, verifying that axioms are satisfied, standard "uniqueness" proofs, proof by contradiction, and disproof by counterexample;

(x) some of the common notions of higher mathematics, possibly including permutations, equivalence relations, the Kronecker delta function, canonical forms, and numerical techniques.

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Last updated July 28, 2004 . Comments, questions, suggestions?