Loosely speaking, linear algebra is a branch of mathematics which studies common properties of a algebraic system. This system consists of a set and some notion of an operation, that is the “linear combination" of the elements in the set.
So called vector space turned out to one of the most useful abstraction of this type of algebraic system.
A vector space (or linear space) consists of the following:
a field F of scalars;
a set V of objects, called vectors;
a rule (or operation), called vector addition, which associates with each pair of vectors α, β in V a vector α + β in V, called the sum of α and β, with the following properties
Commutativity, α + β = β + α;
Associativity, α + (β+γ) = (α+β) + γ;
Existence of a unique vector 0 in V, called the zero vector, such that α + 0 = α for all α in V;
Existence of a unique addtive inverse − α in V for each vector α in V such that α + (−α) = 0;
a rule (or operation), called scalar multiplication, which associates with each scalar c in F and vector α in V a vector cα in V, called the product of c and α, in such a way that
1α = α for every α in V;
(c1c2)α = c1(c2α);
c(α+β) = cα + cβ;
(c1+c2)α = c1α + c2α.
When we associate a vector space V with a field F, we say V is a vector space over the field F. It is important note that the vectors in a vector space is much more general than the vectors we encounter in elementary algebra, which is just a special case in our more general vector space. We call it the n-tuple space and we show more examples of other vector spaces below.
The space of m × n matrices, Fm × n. Let F be any field and let m and n be positive integers. Let Fm × n be the set of all m × n matrices over the field F. The sum of two vectors A and B in Fm × n is defined by (A+B)ij = Aij + Bij. The product of a scalar c and the matrix A is defined by (2−4) (cA)ij = cAij.
The space of functions from a set to a field. Let F be any field and let S be any non-empty set. Let V be the set of all functions from the set S into F. The sum of two vectors f and g in V is the vector f + g, i.e., the function from S into F, defined by (f+g)(s) = f(s) + g(s). The product of the scalar c and the function f is the function cf defined by (cf)(s) = cf(s). The preceding examples are special cases of this one. For an n-tuple of elements of F may be regarded as a function from the set S of integers 1, …, n into F. Similarly, an m × n matrix over the field F is a function from the set S of pairs of integers, (i,j), 1 ≤ i ≤ m, 1 ≤ j ≤ n, into the field F. Verification:
Note in this case for condition 1 and 2, our field F would be the same field F where we are taking our set S to and the vector V will be the functions. For condition 3, vector addition:
Since addition in F is commutative, f(s) + g(s) = g(s) + f(s) for each s in S, so the functions f + g and g + f are identical.
Since addition in F is associative, f(s) + [g(s)+h(s)] = [f(s)+g(s)] + h(s) for each s, so f + (g+h) is the same function as (f+g) + h.
The unique zero vector is the zero function which assigns to each element of S the scalar 0 in F.
For each f in V, (−f) is the function which is given by (−f)(s) = − f(s)
Now for condition 4, vector multiplication:
Since 1 exists for field F, we have (1f)(s) = 1f(s) = f(s)
Since multiplication in F is associative, (c1c2)f(s) = c1(c2f(s)) for each s.
Since multiplication distributive over addition in F, c(f+g)(s) = c(f(s)+g(s)) = cf(s) + cg(s) for each s. So c(f+g) is equivalent to cf + cg.
Similarly, (c1+c2)f(s) = c1f(s) + c2f(s) for every s, so (c1+c2)f is equivalent to c1f + c2f.
The space of polynomial functions over a field F. Let F be a field and let V be the set of all functions f from F into F which have a rule of the form f(x) = c0 + c1x + ⋯ + cnxn where c0, c1, …, cn are fixed scalars in F (independent of x ). A function of this type is called a polynomial function on F. Let addition and scalar multiplication be defined as in Example [space_of_functions_from_set_to_field]. The sum of two vectors f and g in V is defined by (f+g)(s) = f(s) + g(s). The product of the scalar c and the function f is the function cf defined by (cf)(s) = cf(s). One must observe here that if f and g are polynomial functions and c is in F, then f + g and cf are again polynomial functions.
The field ℂ of complex numbers may be regarded as a vector space over the field R of real numbers. More generally, let F be the field of real numbers and let V be the set of n-tuples α = (x1,…,xn) where x1, …, xn are complex numbers. Define addition of vectors and scalar multiplication as in Example [n-tuple_space]. If β= (y1,y2,…,yn) with yi in F, the sum of α and β is defined by α + β = (x1+y1,x2+y2,…,xn+yn). The product of a scalar c and vector α is defined by cα = (cx1,cx2,…,cxn). In this way we obtain a vector space over the field R which is quite different from the space ℂn and the space ℝn. Can you verify this?
There are a few simple facts which follow almost immediately from the definition of a vector space, and we proceed to derive these.
If c is a scalar and 0 is the zero vector, then by 3(c) and 4(c) c0 = c(0+0) = c0 + c0. Adding − (c0) and using 3( d), we obtain c0 = 0.
Similarly, for the scalar 0 and any vector α we find that 0α = 0.
If c is a non-zero scalar and α is a vector such that cα = 0, then by [property:zero_vector], c−1(cα) = 0. But c−1(cα) = (c−1c)α = 1α = α hence, α = 0. Thus we see that if c is a scalar and α a vector such that cα = 0, then either c is the zero scalar or α is the zero vector.
If α is any vector in V, then 0 = 0α = (1−1)α = 1α + (−1)α = α + (−1)α from which it follows that (−1)α = − α.
Finally, the associative and commutative properties of vector addition imply that a sum involving a number of vectors is independent of the way in which these vectors are combined and associated. For example, if α1, α2, α3, α4 are vectors in V, then (α1+α2) + (α3+α4) = [α2+(α1+α3)] + α4 and such a sum may be written without confusion as α1 + α2 + α3 + α4.
A vector β in V is said to be a linear combination of the vectors α1, …, αn in V provided there exist scalars c1, …, cn in F such that $$\begin{aligned} \beta & =\mathrm{c}_1 \alpha_1+\cdots+\mathrm{c}_{\mathrm{n}} \alpha_{\mathrm{n}} \\ & =\sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{c}_{\mathrm{i}} \alpha_{\mathrm{i}} . \end{aligned}$$
Other extensions of the associative property of vector addition and the distributive properties 4(c) and 4(d) of scalar multiplication apply to linear combinations: $$\begin{aligned} \sum_{i=1}^n c_i \alpha_i+\sum_{i=1}^n d_i \alpha_i & =\sum_{i=1}^n\left(c_i+d_i\right) \alpha_i \\ c \sum_{i=1}^n c_i \alpha_i & =\sum_{i=1}^n\left(c c_i\right) \alpha_{i \cdot} \end{aligned}$$