# BDIC1014J Linear Algebra (Engineering)

Linear algebra, a branch of mathematics concerning vectors and linear mappings between vector spaces, is a fundamental course for college mathematics. In this module basic ideas of Linear Algebra will be introduced. Topics covered are: vectors in 2- and 3-dimensional space; dot product, vector product and scalar triple product; orthogonal projection; lines and planes in 3-dimensional space; systems of linear equations; elementary row operations and Gaussian elimination; matrices; matrix algebra; determinants; inverses of matrices; eigenvalue and eigenvectors; diagonalization of a matrix; quadratic forms; vector space; rank of a matrix.

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Curricular information is subject to change

Learning Outcomes:

Students are expected to achieve the following competence:
• Giving algebraic and geometrical representations for vectors
• Performing arithmetic with vectors;
• Using vectors to solve classical geometric problems;
• Manipulating dot and cross products, and using them to solve geometric problems;
• Solving systems of linear equations in terms of Gaussian elimination and Cramer’s rule;
• Performing matrix arithmetic;
• Calculating inverse of a matrix and a determinant;
• Solving eigenvalue problem for a matrix;
• Diagonalizing a matrix;
• Solving a quadratic form problem in terms of matrix diagonalization;
• Being able to identifying the rank of a matrix;
• Giving the space spanned by a number of vectors.

Indicative Module Content:

Teaching Plan

Week 1: Appendix: Revisit to high school mathematics: Set theory, summation and product notations, mathematical induction.
Definitions & properties of vectors.
Week 2: Definitions & properties of vectors, scalar multiplication, linear independence. Tutorial 1.
Week 3: Dot product (algebraic & geometric definitions), projection of vectors, two important inequalities, applications.
Week 4: Cross product (algebraic & geometric definitions), applications. Tutorial 2.
Week 5: Applications of vectors: equations of lines and planes, scalar triple product.
Week 6: Matrices and systems of linear equations, Gaussian elimination. Tutorial 3.
Week 7: Row echelon form, reduced echelon form, homogeneous systems.
Week 8: Revision for mid-term exam. Tutorial 4.
Week 9: Matrix algebra, elementary matrices, partitioned matrices, LU decomposition.
Week10: Determinant of a matrix, properties of determinants. Tutorial 5.
Week12: Definitions of eigenvalue and eigenvector of a matrix. Tutorial 6.
Week13: Diagonalization of matrix.
Week14: Quadratic forms, positive and negative definite matrices, rank of matrix. Tutorial 7.
Week15: Vector spaces and subspaces spanned by vectors.
Week16: Revision for final exam.

Remarks:
1. In each odd week there are two lectures; in each even week there is one lecture and one tutorial.
2. The chapter of vector spaces is only a brief review of the theory. Interested students are encouraged to read reference textbooks, and wait for future advanced courses on algebra theory.

Student Effort Hours:
Student Effort Type Hours
Lectures

64

Practical

66

Total

130

Approaches to Teaching and Learning:
Lecturing (face-to-face teaching) + tutorials + group project on open questions + oral defense in full English + close-book examinations.
Requirements, Exclusions and Recommendations

Not applicable to this module.

Module Requisites and Incompatibles
This module is delivered overseas and is not available to students based at the UCD Belfield or UCD Blackrock campuses

Assessment Strategy
Description Timing Component Scale Must Pass Component % of Final Grade In Module Component Repeat Offered
Exam (In-person): 2 hour final exam Week 15 Graded Yes

100

Yes

Carry forward of passed components
No

Remediation Type Remediation Timing
In-Module Resit Prior to relevant Programme Exam Board