adv lin algie

a field is a set F with two operations (+, *) such that for all a,b,c in F:
(1) a+b=b+a and a*b=b*a
(2) a+(b+c)=(a+b)+c and a*(b*c)=(a*b)*c
(3) there exist unique elements 0,1 in F such that a+0=0+a=a and a*1=1*a=a
(4) for all a in F and b in f\{0} there exist c,d in F such that a+c=0 and b*d=1
examples: R, Q, Z(mod prime), C, Q(sqrt2)

a vector space, defined over a field F, is a set with two operations (+,*) such that the following conditions hold:
(1) for all a,b in V, a+b=b+a
(2) for all a,b,c, in V, a+(b+c) = (a+b)+c
(3) there exists a 0 in V such that 0+x=x+0=x for all x in V
(4) for x in V, there exists a y in V such that x+y=y+x=1
(5) for all x in V, 1*x=x*1=x
(6) for a,b in F, x in V, a(bx)=(ab)x
(7) for a in F, x y in V, a(x+y)=ax+ay
(8) for a,b in F, x in V, (a+b)x= ax+bx
examples: F^n, sequences in a F, M(m*m), f(S,F), P(F)

a subset W of a vector space V over a field F is a subspace of V if W is a vector space over F with the operations of addition and scalar multiplication as defined on V.

let V be a vector space with a subset W. Then W is a subspace of V iff the following 3 conditions hold:
(1) 0 in W (or W non-empty)
(2) W is closed under addition
(3) W is closed under scalar multiplication

A matrix A=(Aij) is symetric if A=At where At is the transpose defined At := (Aji)

symmetric matrices, square matrices with determinant 0, diagonal matrices, upper triangular matrices, polynomials up to a certain degree n, continuous functions from R to R

any intersection of subspaces in a vector space V is a subspace of V

we define the sum of two non-empty subsets S,T in V as S+T := {s+t: s in S, t in T}

a vector space V is called the direct sum of W1 and W2 if W1 and W1 are subspaces of V such that W1intersectW2={0} and W1+W2=V

Let V be a vectors space and W a subspace of V. The set v+W = { v+w : w in W} is called a coset of W. Any vector v1 in v+W is called a representative of the coset v+W

The following are equivalent:
(1) v1+W=v2+W
(2) v1-v2 is in W
(3) (v1+W) intersect (v2+W) is non empty

Let V be a vector space with W a subspace. We define the quotient vector space as V/W := {v+W : v in V} (set of cosets)
addition: (v1+W) + (v2+W) = (v1+v2)+W
scalar multiplication: r(v+W) = (rv)+W

Let V be a vector space with S a non-empty subset of V. We say V is a linear combination of vectors in S if there exist vectors s1,...,sn in S and scalars a1,...,an in F such that v=a1s1+...ansn

Let S be a non-empty subset of a vector space V. The span of S, denoted span(S) is the set of all linear combinations of the vectors in S. For convenience, we define span(empty set) = 0

The span of any subset S of a vector space V is a subspace of V containing S. Moreover, any subspace of V containing S must also contain the span of S

A subset S of a vector space V generates (or spans) V if span(S)=V

A subset S of a vector space V is called linearly dependent if there exist a finite number of vectors s1,..,sn in S and scalars a1,...,an in F not all zero such that a1s1+,...,+ansn = 0.

a subset of a vector space V is linearly independent if it is not linearly dependent

Let V be a vector space with subsets S1 in S2 in V. If S1 is linearly dependent, then S2 is linearly dependent. Similarly, if S2 is linearly independent, S1 is linearly independent.

Let S be a linearly independent subset of a vector space V, and take v in V\S. Then Sunion{v} is linearly dependent iff v in span(S)

A basis of a vector space V is a linearly independent subset of V that generates V

If a vector space V is generated by a finite set S, then some subset of S is a basis for V, hence V has a finite basis

Let V be a vector space generated by a set S containing exactly n vectors. ALso let T be a linearly independent subset of V containing exactly m vectors. Then m<= n and there exists a subset S1 of S containing exactly n-m vectors such that TunionS1 generates V

a vector space is finite-dimensional if it has a basis consisting of a finite number of vectors. The dimension of V is denoted dim(V) and is equal to the number of vectors every basis of V contains.
A vector space that is not finite-dimensional is called infinite dimensional

Let V be a vector space with dimension n:
(1) any finite generating set for V contains at least n vectors and a generating set of V that contains exactly n vectors is a basis for V
(2) any linearly independent subest of V that contains exactly n vectors is a basis for V
(3) Every lineraly independent subset of V can be extended to a basis for V, that is, if T is a linearly independent subset of V, then there is a basis B of V such that T is a subset of B

Let W be a subspace of a finite-dimensional vector space V. Then W is finite-dimensional and dim(W) <= dim(V). Moreover, if dim(W)=dim(V) then W=V

let f be a family of sets. a member M of f is maximal if M is contained in no member of f other than itself

A collection of sets C is called a chain (or nest or tower) if for each pair of sets A and B in c, either A in B or B in A

Let f be a family of sets. If, for each chain C in f there exists a member of f that contains all the members of C, then f contains a maximal member

Let S be a subset of a vector space . A maximal linearly independent subset B of S satisfying:
(1) B is linearly independent
(2) the only linearly independent subset of S that contains B is B itself

Let V be a vector space and S a subset that generates V. If B is a maximal linearly independent subset of S, then B is a basis for V.

Let S be a linearly independent subset of a vector space V. There exists a maximal linearly independent subset of V that contains S.
Corollary: every vector has a basis

Let V and W be vector spaces over the same field F. We call a function T: V->W a linear transformation from V to W if, for all x,y in V and C in F, we have
(1) T(x+y) = T(x) +T(y)
(2) T(cx) = cT(x)

Let V and W be vector spaces and T:V->W be linear. The null space N(T) of T is N(T)= {v in V: T(v) =0}

The range R(T) of T is R(T)={T(v): v in V}

Let V and W be vector spaces and T:V->W be linear. Then N(T) and R(T) are subspaces of V and W respectively

Let V and W be vector spaces and T:V->W be linear. If B={v1,...,vn} is a basis for V, then R(T)=span(T(B))

Let V and W be vector spaces and T:V->W be linear. If N(T) and R(T) are finite-dimensional, then we define the nullity and rank of T as follows:
nullity(T) = dim(N(T))
rank(T) = dim(R(T))

Let V and W be vector spaces and T:V->W be linear. If V is finite dimensional, then nullity(T) + Rank(T) = dim(V)

Let V and W be vector spaces and T:V->W be linear. Then T is injective iff N(T) = {0}

Let V and W be finite dimensional vector spaces and T:V->W be linear. if dim(V)=dim(W), TFAE:
(1) T is one-to-one
(2) T is onto
(3) rank(T) = dim(V)

Let V and W be vector spaces over F and suppose that {v1,..,vn} is a basis for V. For any {w1,...,wn} in W, there exists a unique linear transformation T:V->W such that T(vi)=wi for all 1,...,n

Let V be a finite dimensional vector space. An ordered basis for V is a basis for V endowed with a specific order.

Let B={u1,...,un} be an ordered basis for a vector space V. For x in V there exists unique scalars a1,...,an in F such that x=a1u1+...+anun. we define the coordinate vector of x relative to B to be [x]B = (ai....an) in F^n

Let V and W be finite-dimensional vectors spaces with ordered bases B and C respectively, and let T,U:V-> be linear transformations. Then
(1) [T+U]cb = [T]cb + [U]cb
(2) [aT]cb = a[T]cb for all a in F

Let V,W and Z be vector spaces over the same field F, and let T:V->W and U:V->Z be linear. Then UT (the composition of T and U) is linear

Let V be a vector space and T, U1, U2 in L(V). Then
(1) T(U1+U2) = TU1 + TU2 and (U1+U2)T = U1T+U2T
(2) T(U1U2) = (TU1)U2
(3) Ti = T= iT
(4) a(U1U2) = (aU1)U2=U1(aU2)

Let A,B,C,D,E be matrices with appropriate size
(1) A(B+C) = AB+AC and (D+E)A = DA+EA
(2) a(AB) = (aA)B =A(aB)
(3) Ai = A = iA

Let V and W be finite dimensional vector spaces with ordered bases a and b respectively. Let T:V->W. Then for each v in V we have [T(v)]a = [T]ba[v]b

Let V and W be vector spaces and let T:V->W be linear. A function U:W->V is an inverse of T if TU = Iw and UT=iv. If T has an inverse, we say T is invertible.

The inverse of an invertible linear function is linear.
if T:V->W is invertible, then if either V or W is finite dimensional, then dim(V)=dim(W)

Let A be an nxn matrix. A is invertibel if there exists nxn matrix B such that AB = In = BA

a linear transformation T is invertible iff [T]ba is invertible.
furthermore, [T(-1)]ba = ([T]ba)-1
specifically, A is invertible iff LA is invertible and (LA)(-1) =L(A(-1))

Let V and W be vector spaces. We say that V is isomorphic to W if there exists a linear transformation T:V->W that is invertible.

Let V and W be finite dimensional vector spaces. THen V is isomorphic to W iff dim(V)=dim(W)

Let b be an ordered basis for an n-dimensional vector space V over the field F. The standard representation of V with respect to b is the function phib:V->F^n defined by phib(v)=[v]b. This is an isomorphism

Let b1 and b2 be two ordered bases for a finite-dimensional vector space V< and let Q=[iv]b1b2. Then Q is invertible and for any v in V, [v]b1 = Q[v]b2

Let A,B be nxn matrices. We say that B is similar to A if there exists an invertible matrix Q such that B=Q(-1)AQ

For a vector space V over F, we define the dual space of V to be the vector space L(V,F) denoted by V*. (These functions are linear functionals on V)

Let A be an mxm matrix. Any one of the following three operations of A is called an elementary row(column) operation:
(1) interchanging any two rows of A
(2) multiplying and row by a nonzero scalar
(3) adding any scalar multiple of a row of A to another row

an nxn elementary matrix is a matrix obtained by performing an elementary operation on In. We say it is of type 1,2, or 3 if the elementary operation performed on it is a type 1,2,or 3 respectively

elementary matrices are invertible and the inverse of an elementary matrix is an elementary matrix of the same type

we define the rank of a matrix A to be the rank of the linear transformation LA:F^n to F^n

Let A be mxn, P be mxm and Q be nxn with P and Q invertible. Then rank(AQ)=rank(A)=rank(PA). Therefore rank(PAQ)=rank(A)

elementary operations on a matrix are rank-preserving

The rank of any matrix equals the maximum number of its linearly independent columns, that is the rank of a matrix is the dimension of the subspace generated by its columns.

A system Ax=b of m linear equations in n unknowns is said to be homogenous if b=0. otherwise the system is nonhomogeneous

Let K be the solution set of a consisten system of linear equations Ax=b and let KH be the solution set of the corresponding homogeneous solution. Then for any s in KH, K=s+KH

Let Ax=b be a system of n linear equations in n unknowns. A is invertible iff Ax=b has one solution, namely A(-1)b

Let Ax=b be a system of linear equations. Then the system is consistent iff rank(A)=rank(A|b).

two systems of linear equations are equivalent if they have the same solution set

Let Ax=b be a system of m linear equations in n unknowns and let C be an invertible mxm matrix. then (CA)x=Cb is equivalent to Ax=b

Let Ax=b be a system of m linear equations in n unknowns. If (A1|b1) is obtained from (A|b) by a finite number of elementary row operations, then the system A1x=b1 is equivalent to the original