MATH348 - Advanced Engineering Mathematics 1
E. Kreyszig, Advanced Engineering Mathematics, 9th ed. Section 7.4, pgs. 296-302
Lecture: Linear Independence and Vector Spaces Module: 05
Suggested Problem Set: Suggested Problems : {2, 4, 6, 7, 14, 20}January 26, 2009
E. Kreyszig, Advanced Engineering Mathematics, 9th ed. Section 7.9, pgs. 308-315
Lecture: Abstract V.S. and Inner Product Spaces Module: 06
Suggested Problem Set: Suggested Problems : {8, 9, 10, 27, 29 }January 26, 2009
Quote of Lecture 5
Springfield Scientist: Let’s not listen.
The Simpsons: Bye Bye Nerdie (2001)
Again, we study the problem Ax =bby asking the question, given Aand b, does a solution exist and is this
solution unique. In general the linear system can be solved explicitly using the algorithm of row-reduction.
For the case where there exists a unique solution to the system there are many things, which can be said.
1. Ais an invertible matrix
2. A∼I
3. Ahas npivot positions
4. Ax =0has only the trivial solution
5. Ax =bhas the unique solution x=A−1bfor all b∈Rn
The previous statements are equivalent. That is, they are all true or all false. In the case that Ax =bdoes
not admit a unique solution then they are all false. How can we characterize the problem in this case?
At this point it makes sense to introduce some of the more abstract concepts in linear algebra. The idea
is that if we have unique solutions then we have as many pivots as columns and if we don’t have unique
solutions1then we can expect that pivots6=number-of-columns. Thus it makes sense to determine, given a set
of vectors, which are ‘important.’ We say that the ‘important vectors’ are the linearly independent vectors
of that set and more importantly any vector in the set can be constructed using the linearly independent ones.
Using this concept we then approach the problem in the following way:
1. Given a set of vectors, say the columns of a coefficient matrix, determine which are linearly indepen-
dent.
2. Construct a space of vectors, called the column space of A, defined by the set of all linear combinations
of the linearly independent vectors.
3. Ask if the vector bis in this space. If it is then Ax =bhas a solution. If not, then no solution exists.
This approach will define some new concepts whose vocabulary is listed below:
•Linear Combination
•Linear Independence
•Rank
•Vector Space
1Notice that this would be either no solutions or infinitely many solutions.
