VECTOR SPACES: FIRST EXAMPLES PIETER HOFSTRA 1. Definition So far in the course, we have been studying the spaces Rn and their subspaces, bases, and so on. In mathematics, it is common practice to ask what features of the objects you study are essential for the theory to work, and which are coincidental. We then separate the essential from the inessential, and leap into abstraction: we turn the essential features into a definition, so that we can develop the theory on a more general level. In the case of linear algebra, it turns out that the essential features that make things work for Rn are the fact that you can add vectors and that you can do scalar multiplication (and that they behave properly). For example, the notions of linear combination, linear independence, and basis only refer to addition and scalar multiplication. This leads to the following definition: Definition 1.1 (Abstract Vector Space). A vector space consists of three things: • A set V (to whose elements we sometimes refer as vectors). • An addition operation V × V ! V , written, for v; w 2 V , as (v; w) 7! v + w. • A scalar multiplication operation R × V ! V , written, for v 2 V; c 2 R, as (c; v) 7! cv. These operations are required to satisfy the following axioms: (1) v + w = w + v for all v; w 2 V (commutativity of + ) (2)( v + w) + z = v + (w + z) for all v; w; z 2 V (associativity of + ) (3) There is an element 0 2 V such that 0 + v = v = v + 0 for all v 2 V (4) For all v 2 V there is a unique w with v + w = 0 (5)( cd)v = c(dv) for all c; d 2 R and v 2 V (6)1 v = v for all v 2 V (7)( c + d)v = cv + dv for all c; d 2 R and v 2 V (distributivity 1) (8) c(v + w) = cv + cw for all c 2 R and v; w 2 V (distributivity 2) 2. Examples Of course, the motivating example of a vector space is Rn, with the usual addition of vectors and scalar multiplication. However, there are many other examples. Several of those will be closely related to Rn, but especially the last one (function spaces) is quite different in nature. Example 2.1. Let V be C, the set of complex numbers. When discussing the algebra of complex numbers, we noted that we could define addition of complex numbers by (a + bi) + (c + di) = (a + c) + (b + d)i: Also, given a complex number z = a + bi and a scalar (real number) c, we can multiply them to get cz = c(a + bi) = (ca) + (cb)i: With these notions of addition and scalar multiplication, it is easily verified that C forms a vector space. When we define complex numbers, we make use of the real numbers: given z = a + bi, the two parts a and b are just real numbers. That is, a complex number z = a + bi is completely determined by the two real numbers a and b, and conversely, any two real numbers a and b uniquely determine a complex number a + bi. We might say that C is like R2, but with a different notation: in R2 we write elements as (a; b), while in C we write them as a + bi. Technically, however, they are different sets; in mathematics, 1 2 PIETER HOFSTRA such a situation is made precise by saying that the two sets, while different, are in one-one correspondence with each other. Schematically: C / R2 a + bi / (a; b) But more is true: addition of complex numbers (a + bi) + (c + di) agrees with that of addition of vectors in R2. On the one hand we have (a + bi) + (c + di) = (a + c) + (b + d)i, while on the other hand we have (a; b) + (c; d) = (a + c; b + d). This tells us that not only can we \go back and forth" between the complex numbers and R2, but this passage respects the addition: first adding two complex numbers and then turning it into a vector in R2 gives the same result as first turning both complex numbers into vectors in R2 and then adding them there. Diagrammatically: (a + bi) ; (b + di) / (a; b) ; (c; d) _ _ add individually add (a + c) + (b + d)i / (a + c; b + d) The same holds for scalar multiplication: given z = a + bi and a scalar c, first turning z into (a; b) 2 R2 and then forming c(a; b) = (ca; cb) gives the same result as first doing scalar multiplication in C and then turning the result into a vector in R2: c ; (a + bi) / c ; (a; b) _ _ multiply multiply (ca) + (cb)i / (ca; cb) We may summarize all this informally by saying that not only is C just R2 in disguise as a set, but C is also R2 in disguise as a vector space! p Example 2.2. A polynomial is an expression like p(x) = 2x2 − x + 3, or q(x) = x4 − 5, or r(x) = 1 3 2 x − x + 5. Formally: Definition 2.3. A polynomial (of degree n) is an expression of the form n n−1 anx + an−1x + ··· + a1x + a0 where x is a variable and where the a0; : : : ; an are real numbers, called the coefficients of the polynomial. n n Given two polynomials p(x) = anx + ··· + a1x + a0 and q(x) = bnx + ··· + b1x + b0, we have p(x) = q(x) precisely when ai = bi for all i = 0; : : : ; n. The set of all polynomials of degree n is denoted Pn[x], and the set of all polynomials (of arbitrary degree) P[x]. We now explain how to add polynomials and how to define scalar multiplication (although you already n n know this from calculus!). Given two polynomials p(x) = anx + ··· + a1x + a0 and q(x) = bnx + ··· + b1x + b0, we define p(x) + q(x) = (an + bn)x + ··· + (a1 + b1)x + (a0 + b0): That is, we add polynomials by adding their coefficients. n For scalar multiplication, consider Given two polynomials p(x) = anx + ··· + a1x + a0 and c 2 R, and define n cp(x) = (can)x + ··· + (ca1)x + (ca0): With these definitions, Pn[x] becomes a vector space. Moreover, using the fact that we can always regard a polynomial of degree n as one of higher degree by adding coefficients of zero, we also get that P[x] is a vector space as well. n Just as with complex numbers, a polynomial p(x) = anx + ··· a1x + a0 is completely determined by n+1 its coefficients (an; : : : ; a0), which form an element of R . Moreover, addition and scalar multiplication n+1 are defined coefficient-wise; this means that Pn[x] is just R in disguise! VECTOR SPACES: FIRST EXAMPLES 3 Example 2.4. Matrices. We know from matrix algebra that matrices of the same size can be added entry-wise. Also, we can multiply any matrix by a scalar. This means that when we fix m; n, the set Mm;n of all mxn matrices is a vector space. Since an mxn matrix can be regarded as n column vectors in Rm (or, of course, as m row vectors in n mn R ), we find that Mm;n is R in disguise. For example, we can translate a 3x2 matrix to a vector in R6 as follows: a b c 7−! (a; b; c; d; e; f) 2 6 d e f R Example 2.5. Our last example is different from the previous ones, in the sense that it is not a disguised form of Rn. Consider F[R], the set of all functions from the reals to the reals. So an element of F[R] is a function f : R ! R. (Remember, a function has to give a unique output f(x) for every input x, so something like f(x) = 1=x is not an element of F[R], since it is not defined for x = 0.) From calculus, we know that it is possible to add functions (pointwise, as it is sometimes called): given two functions f; g : R ! R, define a new function f + g : R ! R by (f + g)(x) = f(x) + g(x) Note that we're defining this new function f +g by specifying what it's output is given an arbitrary input value x: to know a function h is to know h(x), for all possible values of x. Note also that this equation uses the + symbol twice: on the left hand side, it means addition of functions (which is what we're trying to define) and on the right hand side it's ordinary addition of real numbers (something we assume we understand already). For scalar multiplication, consider f : R ! R and a real number c 2 R. Define a new function cf : R ! R by (cf)(x) = c · f(x) That is, on input x, the function cf returns the output value c times f(x). With these definitions of addition and scalar multiplication, F[R] is a vector space. Note that the \origin" in this vector space is the constant function z(x) = 0. 3. A bit of linear algebra We now proceed to examine the concepts of linear combinations and linear (in)dependence in the examples given above. Recall first the definition of linear combination: it makes sense in any vector space. Definition 3.1 (Linear Combination). Let V be a vector space, and let v1; : : : ; vk be elements of V .A linear combination of v1; : : : ; vk is a vector v of the form v = a1v1 + ::: + akvk for some real numbers a1; : : : ; ak.