MATH348 - Advanced Engineering Mathematics 1
E. Kreyszig, Advanced Engineering Mathematics, 9th ed. Section 7.8, pgs. 315-322
Lecture: Construction and Properties of A−1Module: 04
Suggested Problem Set: Suggested Problems : {7, 8, 15, 17, 19}June 17, 2009
E. Kreyszig, Advanced Engineering Mathematics, 9th ed. Section 7.7, pgs. 308-315
Lecture: Determinants and Cramer’s Rule Module: 05
Suggested Problem Set: Suggested Problems : {5, 6, 13, 15, 16, 19, 24(a,b,c)}June 17, 2009
Quote of Lecture 4
Jerry: (Knocking on Kramer’s door) Hello? Is Kramer home? Oh, hey.
Kramer: (Spraying his flowers) Hello, neighbour.
Jerry: Boy, those azaleas are really coming in nicely.
Kramer: Oh, you gotta mulch. You’ve got to.
Jerry: You barbecuing tonight?
Kramer: (Ringing his wind chimes) Right after the fireworks.
Seinfeld: The Serenity Now (1997)
Up to this point we have built some logic and intuition supporting the idea that to methodically solve a
linear system efficiently we rewrite the system as a matrix vector product, Ax =b, where Am×nis a matrix
containing the coefficient data for the system and bis an inhomogeneity, which corresponds to translations
of the linear equations in space. The goal now is to find the vector unknown xand to do this we apply the
row-reduction algorithm to the corresponding augmented matrix [A|b]. 1
We say that if m < n then there are fewer equations than unknowns and that the system is under-determined
and expect that at best there may be infinitely many solutions.2If m>nthen there are more equations
than unknowns and the system is over-determined and in this case we expect that solutions may exist and
possibly be unique.3The final case, m=n, provides more intuition about the behavior of solutions to linear
systems. In the following we recap the results for square systems (as many equations as unknowns) and add
to this growing list of equivalent statements:
1. There exists a unique solution, x∈Rn×1to the system Ax =bfor every b∈Rn×1.
2. The homogeneous system Ax =0has only the trivial solution,x=0.
3. A∼I
4. There is a pivot in every column of the coefficient matrix.
5. The columns of Aare linearly indep endent.4
1At this point it should be clear to the reader that if A∈Rm×nthen x∈Rn×1and b∈Rm×1.
2Quick sanity check: If two planes are placed in space then at best they will intersect at a line or be the same
plane.
3See homework 1 problem 5 for a case where three-lines are given and that these three lines have a common
intersection.
4We haven’t yet used the term ‘linearly independent’, but we will.
