MH1201 - LINEAR ALGEBRA IIAY 2014-2015Dr. Le Hai Khoi ([email protected])Division of Mathematical Sciences, SPMS, NTU
PrefaceThese lecture notes are to be used together with the textbook  and the reference-book. During the lectures we will follow these notes carefully. However, you might find thesenotes to be too brief and lacking in examples. The textbooks on the other hand have plentyof examples and exercises, so when you find these notes lacking, you can try to look up thecorresponding material in the textbooks.For several theorems and their proofs, we havebenefited from some different sources other than the textbooks.You should also know that these notes will be updated continuously during the courseas we will surely find errors along the way. We would be very thankful if you could reportany mistakes you find to us so they can be corrected as quickly as possible.We are now ready to start investigating the exciting topic of Linear Algebra II.3
Chapter 1Vector SpacesIn MH1200, we already know how the setRnof alln-tuples of real numbers, together withthe coordinate-wise addition and scalar multiplication operations defined on it, has the samealgebraic properties as the familiar algebra of geometric vectors.The goal of this chapter is to develop a notion of vectors that encompasses not onlyphysical interpretations (forces, velocities, accelerations, ...) and geometric interpretations(vectors), but also the many mathematical applications.1.1Basic NotationLetVbe a non-empty set, whose elements are calledvectors, andRthe set of real numbers,whose elements are calledscalars.Let further, onVtwo operations (calledadditionandscalar multiplication) are definedin the following sense:- Vectors can be added: we use the usual sign + to denote an addition operation, andthe result of adding two vectorsu, v∈Vis denoted byu+v.- Vectors can be multiplied by scalars: we use the usual notationλuto denote the resultof scalar multiplying the vectoru∈Vby the scalarλ∈R.Note that the vector addition and/or the scalar multiplication, in general, will not yieldanother vector inV. For example, ifV=Z, the set of all integers, then the usual operationsof addition and scalar multiplication define addition and scalar multiplication operations onV. However,λuis not always inV.