Linear algebra is the study ofvector spacesandlinear transformations. It is not simply the study of matrices, although matrix theory takes up most of linear algebra.
It is common in mathematics to consider abstraction, which is simply a means of talking about more than one thing at the same time. A vector spaceV is an abstract algebraic structure defined using axioms. There are many examples of vector spaces, such as the sets of real or complex numbers themselves, the set of all polynomials, the set of row or column vectors of a given dimension, the set of all infinite sequences of real or complex numbers, the set of all matrices of a given size, and so on. The beauty of an abstract approach is that we can talk about all of these, and much more, all at once, without being specific about which example we mean.
A vector space is a set whose members are called vectors, on which there are two algebraic operations, calledscalar multiplication andvector addition. As in any axiomatic approach, these notions are intentionally abstract. A vector is defined to be a member of a vector space, nothing more. Scalars are a bit more concrete, in that scalars are almost always real or complex numbers, although sometimes, but not in this book, they are members of an unspecified finite field. The operations themselves are not explicitly defined, except to say that they behave according to certain axioms, such as associativity and distributivity.
Ifv is a member of a vector space V andαis a scalar, then we denote byαvthe scalar multiplication ofv byα. Ifwis also a member ofV, then we denote byv+wthe vector addition ofvandw. The following properties serve to define a vector space, withu,v, andwdenoting arbitrary members ofV andαandβ arbitrary scalars:
• 1.v+w=w+v;
• 2.u+ (v+w) = (u+v) +w;
• 3. there is a unique “zero vector” , denoted 0, such that, for everyv,
v+ 0 =v;
• 4. for eachv there is a unique vector−v such thatv+ (−v) = 0;
• 5. 1v=v, for allv;
• 6. (αβ)v=α(βv);
• 7.α(v+w) =αv+αw;
Ex. 3.1 Show that, if z+z=z, thenz is the zero vector.
Ex. 3.2 Prove that 0v = 0, for all v ∈ V, and use this to prove that
(−1)v=−v for allv∈V. Hint: use Exercise 3.1.
We then write
w−v=w+ (−v) =w+ (−1)v,
for allv andw.
Ifu1, ..., uN are members ofV andc1, ..., cN are scalars, then the vector
x=c1u1+c2u2+...+cNuN
is called a linear combination of the vectors u1, ..., uN, with coefficients
c1, ..., cN.
If W is a subset of a vector space V, then W is called a subspace of
V if W is also a vector space for the same operations. What this means is simply that when we perform scalar multiplication on a vector in W, or when we add vectors in W, we always get members of W back again. Another way to say this is thatW isclosed to linear combinations.
When we speak of subspaces of V we do not mean to exclude the case ofW =V. Note that V is itself a subspace, but not a proper subspace of
V. Every subspace must contain the zero vector, 0; the smallest subspace ofV is the subspace containing only the zero vector,W ={0}.
Ex. 3.3 Show that, in the vector space V =R2, the subset of all vectors
whose entries sum to zero is a subspace, but the subset of all vectors whose entries sum to one is not a subspace.
Ex. 3.4 Let V be a vector space, andW andY subspaces ofV. Show that the union of W and Y, written W ∪Y, is also a subspace if and only if eitherW ⊆Y orY ⊆W.
We often refer to things like
1 2 0
as vectors, although they are but one example of a certain type of vector. For clarity, in this book we shall call such an object areal row vector of dimension threeor areal row three-vector. Similarly, we shall call
3i −1 2 +i 6
acomplex column vector of dimension four
or acomplex column four-vector. For notational convenience, whenever we refer to something like a real three-vector or a complex four-vector, we shall always mean that they are columns, rather than rows. The space of real (column) N-vectors will be denoted RN, while the space of complex (column)N vectors isCN.
notion of the size or dimension of the vector space. A vector space can be finite dimensional or infinite dimensional. The spacesRN andCN have dimensionN; not a big surprise. The vector spaces of all infinite sequences of real or complex numbers are infinite dimensional, as is the vector space of all real or complex polynomials. If we choose to go down the path of finite dimensionality, we very quickly find ourselves talking about matrices. If we go down the path of infinite dimensionality, we quickly begin to discuss convergence of infinite sequences and sums, and find that we need to introduce norms, which takes us into functional analysis and the study of Hilbert and Banach spaces. In this course we shall consider only the finite dimensional vector spaces, which means that we shall be talking mainly about matrices.