MATH348 - Advanced Engineering Mathematics 1
E. Kreyszig, Advanced Engineering Mathematics, 8th ed. Section 7.1, pgs. 272-278
Lecture: Matrices, Vectors: Algebra of ‘+’ Module: 01
Suggested Problem Set: Suggested Problems : {5, 7 }August 25, 2008
E. Kreyszig, Advanced Engineering Mathematics, 8th ed. Section 7.2, pgs. 278-287
Lecture: Matrix Multiplication : Algebra of ‘·’ Module: 01
Suggested Problem Set: Suggested Problems : {3, 5, 8, 13, 19, 20, 22 }August 25, 2008
Quote of Lecture 1
Remember where the thought is. I brought all this so you can survive when law is lawless.
Gorillaz: Clint Eastwood (2001)
We begin the course with the algebra of matrices. To do this we must do the following:
•Define a mathematical object called a matrix.
•Define how groups of matrices behave with respect to the binary operations + and ·.
once this is completed we can begin to study how this algebra relates to linear systems and how we should
think about solutions to linear problems. You have studied linear equations before in the sense that you
have asked the question, for what values of xdoes the equation,
ax =b, a, b ∈R,(1)
have a solution? It should be straightforward to see that for a6= 0 the problem has the unique solution,
x=b
a, a 6= 0.(2)
What about the case where a= 0? Well, that is trickier. If b= 0 then the value of xdoesn’t matter. We
always have equality, there are infinitely many choice for x! However, in the case where b6= 0 we have the
inconsistent equation, 0 cot x=b6= 0. There are no values of x, which will satisfy the equation.
However, we are getting ahead of ourselves. The question we need to ask now is given a lot of data how
can we systematically organize it? Also, once this is done how can we manipulate groups of them? Without
answers to these questions we have no hope of setting up equations (1)-(2) for large data. In what follows
will record the definitions of matrices and their algebraic structure.1
1From the perspective of abstract algebra, a field that studies collections of objects and their properties with resp ect
to binary operators, we would call this sort of structure a non-commutative ring.
