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101 Cards in this Set

  • Front
  • Back
Ring
a ring R is a nonempty set with 2 binary operations such that
1 a + b = b + a
2 ( a + b ) + c = a + ( b + c)
3 there is an additive identity 0.
4 every element has an additive inverse
5 a(bc) = (ab)c
a(b+c)=ab+acand (b+c)a=ba+ca
subring
a nonempty subset S of a ring R is a subring if S is closed under subtraction and multiplication. That is if a and b are in S then (a-b) and ab are in S
Integral Domain
a commutative ring with unity and no zero divisors (=cancellation law holds)
Zero Divisor
nonzero element of a commutative R such that there is a nonzero element b and ab=0
field
commutative ring with unity in which every nonzero element is a unit
Ideal
A subring I of a ring R with the following property: if a is in I and r is in R, then ar and ra are both in I.
Factor Ring
R/A = {r + A | r in R}
R?A is a ring iff A is an ideal of R
Define a principal ideal and name the properties of the ring R in which it may occur.
R must be COMMUTATIVE WITH IDENTITY. If c is in R, the principal ideal generated by c is the set of all multiples of c in R.
R/I
is the QUOTIENT RING and is indeed a ring. If R is commutative, so is R/I. If R has identity, so does R/I.
What is true of the kernel of a ring homomorphism that maps R to S?
The kernel is an ideal in R. If the kernel is {0_R} only, then the function is injective.
State the FIRST ISOMORPHISM THEOREM.
Let f:R→S be a surjective homomorphism of rings with kernel K. Then R/K is isomorphic to S.
If R is a ring with identity, what is true of the set of all units in R?
They form a group under multiplication.
Commutative ring
a ring with the added axiom that multiplication is commutative
Ring with identity
a ring that contains a multiplicative identity 1
Rings of functions
Rings whose elements are functions over a set. The operations are defined as function addition and function multiplication.
Subring subfield
A subring is a subset of a ring that satisfies all of the ring axioms under the same operations as defined for the parent ring. The same goes for a subfield, replacing the word ring with field.
Cartesian product
The cartesian product of two sets A and B is the set A X B of all ordered pairs (a,b), where a is in A and b is in B. If A and B are rings A X B is a ring where operations are defined component-wise A X B is commutative if both A and B are and A X B has identity if both A and B do.
Unit
An element a of a ring R WITH IDENTITY is a unit if there exists some element b in R such that ab=1=ba. b is the multiplicative inverse.
Multiplicative Inverse
The multiplicative inverse of the nonzero element a in a ring R WITH IDENTITY is the nonzero element b for which ab=1=ba.
Zero divisor
An element a in a ring R is a zero divisor if:
1- a≠0
2- There exists some b≠0 in R such that ab=0.
Ring isomorphism
A bijective function f between rings that has these properties: f(a+b)=f(a)+f(b)
Ring homomorphism
A function f between rings that has these properties: f(a+b)=f(a)+f(b)
Image of a function f
The set of all elements of the codomain to which f maps an element of the domain.
Extension Field
A field created by including new elements in a previously defined field. The ring F[x]/p(x) may be an extension field of F.
Principal ideal
An ideal in a COMMUTATIVE ring WITH IDENTITY generated by a single element. In other words, the set of all multiples of an element of the ring.
a ≡ b (mod I) a ≡ b (mod I) if a-b is in I
Left coset
a+I The set of all elements b of a ring such that a-b is in I.
Homomorphic image
S is a homomorphic image of R if f:R→S is a surjective homomorphism of rings
subring
a nonempty subset S of a ring R is a subring if S is closed under subtraction and multiplication. That is if a and b are in S then (a-b) and ab are in S
True or false finite integral domains are fields
True`
t or f Z_p is a field
true
Theorem 14.4 R/A is a Field iff A is Maximal
Let R be a communitive rind with unity and let A be an ideal or R. Then R/A is a field iff A is maximal.
The ten axioms to verify Vector Spaces? (u
v,w are vectors m,k are scalars W and V are vector spaces),1.) u +v is in W
2.) u +v = v + u
3.) u +(v +w)= (u +v) + w
4.) There is a 0 so that u + 0 = o +u =u.
5.) there is a -u so that u =(-u)=0
6.)ku is in V
7.) k(u+v)=vu +kv
8.) (k +m)u=ku +mu
9.) k(mu)=-(km) u
10.) 1u =u
Subspace
W is a subspace of V is W is a vector space itself and the same rules of addition and multiplication apply in W as they do in V
Span of S
The subspace of V that is formed from all possible linear combinations of a non-empty set S is called the span of S and is noted span{w1...w} or Span(S)
Basis of a Vector Space
S is a basis of V if :
a.) S is linearly independent
b.) S spans V
Checking if a set is a Basis
set Ax=0 and Ax=b. The matrix of coeffecients will be the same for the two. Theorem 2.3.8 says that if the det≠0 then both systems have the solutions we want and therefore it is a basis.
Coordinate Vector
Let S={v1,v2....vn} be a basis for vector space V. v=c1v1 +c2v2....+cnvn is the expression for the vector v, then [v]s={c1,c2...cn} is called the coordinate vector of v relative to S.
Dimension
The dimension of a finite-dimensional vectorspace V is denotes as dim(V) and represent the # of vectors found in a basis for that vector space.
Transition matrix
The matrix that, when multiplied on the right by a coordinate vector relative to a basis B returns the coordinate vector relative to basis S.
How to find the transition matrix
PB to Bⁱ is found by forming the matrix [Bⁱ|B] and doing gauss Jordan until you end up with [I|PB to Bⁱ],[new basis|old basis]
Cross Product
|v||u| sin (ø),Only valid in R3 and the cross product is orthogonal to both original vectors.
Orthogonality
Two vectors are orthogonal if the result of their dot product is zero.
Fundamental estimate of linear algebra
No linearly independent subset of a given vector space has more elements than a generating set. Thus if V is a vector space L is contained in V a linearly independent set and E is contained in V a generating set then the |L|<=|E|
Steinitz Exchange Theorem
Lets you compare two sets. Let V be a vector space and L is contained in V finite linearly independent subset and E is contained in v for a generating set. Then there is an injection phi:L->E such that (E\phi(l)) U L is also a generating set for V
Exchange Lemma
Let V be a vector space with E is contained in M a generating set for V containing a linearly independent subset
Basis
A linearly independent spanning set
Dimension
number of elements in the basis
Cardinality Criterion of Bases
Let V be a finitely generated vector space
(1) each linearly independent subset L contained in B has at most dim V elements and if |L|= dim V then L is actually a basis
(2) Each generating set E contained in V has at least dim V elements and if |E| = dim V then E is a basis.
Dimension Estimate for Vector Subspaces
If the dimension is the same and its a subspace it isn't a proper subspace.
Linear Mapping
V W be vector spaces over a field f. A mapping f:V->W is called a linear on more precisely F-linear or even a homomorphism of F vector spaces if for all v1,v2 which are members of V and f in F we have
f(v1+v2)=f(v1)+f(v2)
af(v)=f(av)
Rank-Nullity Theorem
Rank - im (f)
nullity- dim (ker f)
f:V->W is linear
dim(im f)+dim(ker f) =dim V
Layman's steinitz
we can swap same elements of our linearly independent set and still keep a generating set
The Steinitz exchange lemma is a basic theorem in linear algebra used, for example, to show that any two bases for a finite-dimensional vector space have the same number of elements
Cardinality criterion of bases
Let V be a finitely generated vector space
(1) Each linearly independent subset L is contained in B has at most dim V elements, and if |L|=dim V then L is a basis
(2) Each generating set E is contained in V has at least dim V elements and if |E|=dim V then E is a basis
Cardinality theorems of set
|L|<|B|<|E| where L is a linearly independent subset B is a Basis E is a generating set
Dimension estimate for vector subspaces
A proper subspace of a finitely dimensional vector space has itself a strictly smaller dimension
Dimension Theorem - Let V be a vector space containing vector subspaces U
W is in V
dim(u+w)+dim(u n w)=dim U+dimV
isomorphism of vector spaces
A bijective linear mapping is called an isomorphism of vector spaces. If there is an isomorphism between two vector spaces we call them isomorphic. A homomorphism from one vector space to itself is called an endomorphism of our vector space.
automorphism of a vector space
an isomorphism of a vector space to itself is called an automorphism of a vector space.
Fixed point
a point that is sent to itself by a mapping is called a fixed point. Given a mapping f:X->X we denote the set of fixed points by
X^f={x is a member of X:f(x)=x}
Complimentary
two vector subspaces V1,v2 of a vector space V are called complimentary if addition defines a bijection V1 x V2 ~>V
The classification vector spaces by their dimension
let n be a natural number. Then a vector space over a field F is isomorphic to F^n iff it has dim n
Cancellation theorem
if R is an integral domain and ab=ac then either a=0 or b=c
Finite integral domain
a field
monic
P in R[x] is monic if its leading coefficient is one
Division and remainder theorem
Let R be a non zero commutative ring and let P Q in R[x] with Q monic. then there exists Ab in R[x]|p=AQ+B and deg(B)<deg(q) or B=0
Null ring
Here R is a single element set, say {0} with the operations 0+0=0 and 0x0=0. I'll call any ring that is not the zero ring a nonzero ring
Algebraically closed
A field F is algebraically closed if each non constant polynomial P in F[x]\F with coefficients in our field has a root in our field F
Fundamental Theorem of Algebra
The field of complex numbers C is algebraically closed
Determinant
R is a commutative ring A is a real nxn matrix det (A)=sum over each length of permutation of sgn(length of perm)all(1)a2l(2)...anl(n)
cramer's rule
A. adj A =det (A) (I)
a matrix is invertible iff
its determinant is a unit in R
an endomorphism over a non zero finite dimensional vector space has an eigenvalue if
it is over an algebraically closed field
Perron 1907 thm
If M is a Markov matrix all of whose entries are positive then the eigenspace E(1,M) is one dimensional. There exists a unique basis vector all of whose entries are positive real numbers such that the sum of its entries is 1.
inner product satisfies the following axioms
(1) scalar multiplication and addition (ax+by,z) = a(x,z)+b(y,z)
(2) symmetry (x,y) = (y,x)
(3) bigger than 0 (x,x)>=0 with equality iff x=0
a complex inner product satisfies the following axioms
(1) addition and multiplication (ax+by,z) = a(x,z)+b(y,z)
(2) symmetry (v,w)=(w,v)*
(3) bigger than 0 0 (x,x)>=0 with equality iff x=0
orthonormal sufficient thm
let V be a finite dimensional inner product space. Then V has an orthonormal basis.
norm
||V||=(v,v)^0.5
an invertible 3 x 3 matrix that has trace 0
[ 2 0 0
0 -1 0
0 0 -1]
an infinite dimensional vector space
R[x]
a vector space with exactly 8 elements
F^3_2
a linear mapping f:R^3->R^3 whose rank is 1
f(x,y,z)=(x,0,0)
an injective linear mapping from a vector space of dimension 2 to a vector space of dimension 3
f(x,y)->(x,y,0)
a non zero idempotent mapping f:V->V that isnt the identity
abs
a symmetric bilinear form
dot product
an alternating bilinear form
cross product
an alternating multi linear form on vx v x v ->R for some non zero real vector space V
det of 3 real matrix
an integral domain that is not a field
R[x]
a ring that is not commutative
matrix
an ideal in a ring that is not principal
(2,x) in z[x]
a commutative ring that is not an integral domain
z/4z
a non zero polynomial with more roots than its degree
2(x^2+x) in z/4z[x]
a commutative ring with unity
z
a set with composition
end(v)
a field where every element is a unit
Q
an algebraically closed field
complex numbers
a ring homomorphism
the inclusion z->Q
a ring homomorphism f:R->S such that the identity for r isnt the identity of s
f:R->Mat(2;R) f(x)=[x 0, 0 0] for all x in R
an ideal of a commutative ring
mz in z
principal ideals in a commutative ring
{0} in R,
subring of C
gaussian integers
diagonalisable nilpotent matrix
0 matrix