Some notes on linear algebra Throughout these notes, k denotes a field (often called the scalars in this context). Recall that this means that there are two binary operations on k, denoted + and ·, that (k; +) is an abelian group, · is commutative and associative and distributes over addition, there exists a multiplicative identity 1, and, for all t 2 k, t 6= 0, there exists a multiplicative inverse for −1 t, denoted t . The main example in this course will be k = C, but we shall also consider the cases k = R and k = Q. Another important case is k = Fq, a finite field with q elements, where q = pn is necessarily a prime power (and p is a prime number). For example, for q = p, Fp = Z=pZ. If we drop the requirement that multiplication be commutative, then such a k is called a division algebra or skew field. It is hard to find noncommutative examples of division rings, though. For example, by a theorem of Wedderburn, a finite division ring is a field. One famous example is the quaternions H = R + R · i + R · j + R · k: A typical element of H is then of the form a0 + a1i + a2j + a3k. Here i2 = j2 = k2 = −1, and ij = k = −ji; jk = i = −kj; ki = j = −ik: These rules then force the rules for multiplication for H, and a somewhat lengthy check shows that H is a (noncommutative) division algebra. We shall occasionally relax the condition that every nonzero element has a multiplicative inverse. Thus a ring R is a set with two binary operations + and ·, that (R; +) is an abelian group, · is associative and distributes over addition, and there exists a multiplicative identity 1. If · is commutative as well, we call R a commutative ring. A standard example of a noncommuta- tive ring is the ring Mn(R) of n × n matrices with coefficients in R with the usual operations of matrix addition and multiplication, for n > 1, and with multiplicative identity the n × n identity matrix I. (For n = 1, it is still a ring but in fact it is isomorphic to R and hence is commutative.) More 1 generally, if k is any field, for example k = C and n > 1, then the set Mn(k) of n × n matrices with coefficients in k with the usual operations of matrix addition and multiplication is a noncommutative ring. 1 Definition of a vector space Definition 1.1. A k-vector space or simply a vector space is a triple (V; +; ·), where (V; +) is an abelian group (the vectors), and there is a func- tion k × V ! V (scalar multiplication), whose value at (t; v) is just denoted t · v or tv, such that 1. For all s; t 2 k and v 2 V , s(tv) = (st)v. 2. For all s; t 2 k and v 2 V ,(s + t)v = sv + tv. 3. For all t 2 k and v; w 2 V , t(v + w) = tv + tw. 4. For all v 2 V , 1 · v = v. A vector subspace W of a k-vector space V is a subgroup W of (V; +) such that, for all t 2 k and w 2 W , tw 2 W , i.e. W is a subgroup closed under scalar multiplication. It is then straightforward to check that W is again a k-vector space. For example, f0g and V are always vector subspaces of V . It is a straightforward consequence of the axioms that 0v = 0 for all v 2 V , where the first 0 is the element 0 2 F and the second is 0 2 V , that, for all t 2 F , t0 = 0 (both 0's here are the zero vector), and that, for all v 2 V ,(−1)v = −v. Hence, W is a vector subspace of V () W is nonempty and is closed under addition and scalar multiplication. Example 1.2. (0) The set f0g is a k-vector space. (1) The n-fold Cartesian product kn is a k-vector space, where kn is a group under componentwise addition and scalar multiplication, i.e. (a1; : : : ; an) + (b1; : : : ; bn) = (a1 + b1; : : : ; an + bn) t(a1; : : : ; an) = (ta1; : : : ; tan): (2) If X is a set and kX denotes the group of functions from X to k, then kX is a k-vector space. Here addition is defined pointwise: (f + g)(x) = f(x) + g(x), and similarly for scalar multiplication: (tf)(x) = tf(x). 2 (3) With X a set as in (2), we consider the subset k[X] ⊆ kX of functions f : X ! k such that the set fx 2 X : f(x) 6= 0g is a finite subset of X, then k[X] is easily seen to be a vector space under pointwise addition and scalar multiplication of functions, so that k[X] is a vector subspace of kX . Of course, k[X] = kX if and only if X is finite. In general, for a function f : X ! k, we define the support of f or Supp f by: Supp f = fx 2 X : f(x) 6= 0g: Then k[X] is the subset of kX consisting of all functions whose support is finite. Given x 2 X, let δx 2 k[X] be the function defined by ( 1; if y = x; δx(y) = 0; otherwise: The function δx is often called the delta function at x. Then clearly, for every f 2 k[X], X f(x) = f(x)δx; x2X where in fact the above sum is finite (and therefore meaningful). We will often identify the function δx with the element x 2 X. In this case, we can view X as a subset of k[X], and every element of k[X] can be uniquely P written as x2X tx · x, where the tx 2 k and tx = 0 for all but finitely n many x. The case X = f1; : : : ; ng corresponds to k , and the element δi th to the vector ei = (0;:::; 1;:::; 0) (the i component is 1 and all other components are 0). For a general ring R, the analogue of a vector space is an R-module, with some care if R is not commutative: Definition 1.3. A left R-module M is is a triple (M; +; ·), where (M; +) is an abelian group and there is a function R × M ! M, whose value at (r; m) is just denoted r · m or rm, such that 1. For all r; s 2 R and m 2 M, r(sm) = (rs)m. 2. For all r; s 2 R and m 2 M,(r + s)m = rm + sm. 3. For all r 2 R and m; n 2 M, r(m + n) = rm + rn. 4. For all m 2 M, 1 · m = m. 3 Left submodules of an R-module M are defined in the obvious way. A right R-module is an abelian group M with a multiplication by R on the right, i.e. a function R × M ! M whose value at (r; m) is denoted mr, satisfying (ms)r = m(sr) and the remaining analogues of (2), (3), (4) above. Thus, if R is commutative, there is no difference between left and right R-modules. Example 1.4. (1) The n-fold Cartesian product Rn is a left R-module, where Rn is a group under componentwise addition and scalar multiplica- tion, i.e. (a1; : : : ; an) + (b1; : : : ; bn) = (a1 + b1; : : : ; an + bn) r(a1; : : : ; an) = (ra1; : : : ; ran): Of course, it can also be made into a right R-module. (2) If X is a set and RX denotes the group of functions from X to R, then RX is a left R-module using pointwise addition and scalar multiplication. Similarly the subset R[X], defined in the obvious way, is a submodule. (3) Once we replace fields by rings, there are many more possibilities for R-modules. For example, if I is an ideal in R (or a left ideal if R is not commutative), then I is an R-submodule of R and the quotient ring R=I (in the commutative case, say) is also an R-module. For example, Z=nZ n is a Z-module, but it looks very different from a Z-module of the form Z . For another example, which will be more relevant to us, if R = Mn(k) is the (noncommutative) ring of n × n matrices with coefficients in the field k, then kn is a left R-module, by defining A · v to be the usual multiplication of the matrix A with the vector v (viewed as a column vector). 2 Linear maps Linear maps are the analogue of homomorphisms and isomorphisms: Definition 2.1. Let V1 and V2 be two k-vector spaces and let F : V1 ! V2 be a function (= map). Then F is linear or a linear map if it is a group homomorphism from (V1; +) to (V2; +), i.e. is additive, and satisfies: For all t 2 k and v 2 V1, F (tv) = tF (v). The composition of two linear maps is again linear. For example, for all V1 and V2, the constant function F = 0 is linear. If V1 = V2 = V , then the identity function Id: V ! V is linear. The function F is a linear isomorphism or briefly an isomorphism if it is both linear and a bijection; in this case, it is easy to check that F −1 is 4 ∼ also linear. We say that V1 and V2 are isomorphic, written V1 = V2, if there exists an isomorphism F from V1 to V2.