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101 Cards in this Set
- Front
- Back
Ring
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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 |
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subring
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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
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Integral Domain
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a commutative ring with unity and no zero divisors (=cancellation law holds)
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Zero Divisor
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nonzero element of a commutative R such that there is a nonzero element b and ab=0
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field
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commutative ring with unity in which every nonzero element is a unit
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Ideal
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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.
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Factor Ring
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R/A = {r + A | r in R}
R?A is a ring iff A is an ideal of R |
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Define a principal ideal and name the properties of the ring R in which it may occur.
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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.
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R/I
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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.
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What is true of the kernel of a ring homomorphism that maps R to S?
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The kernel is an ideal in R. If the kernel is {0_R} only, then the function is injective.
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State the FIRST ISOMORPHISM THEOREM.
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Let f:R→S be a surjective homomorphism of rings with kernel K. Then R/K is isomorphic to S.
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If R is a ring with identity, what is true of the set of all units in R?
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They form a group under multiplication.
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Commutative ring
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a ring with the added axiom that multiplication is commutative
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Ring with identity
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a ring that contains a multiplicative identity 1
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Rings of functions
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Rings whose elements are functions over a set. The operations are defined as function addition and function multiplication.
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Subring subfield
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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.
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Cartesian product
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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.
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Unit
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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.
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Multiplicative Inverse
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The multiplicative inverse of the nonzero element a in a ring R WITH IDENTITY is the nonzero element b for which ab=1=ba.
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Zero divisor
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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. |
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Ring isomorphism
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A bijective function f between rings that has these properties: f(a+b)=f(a)+f(b)
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Ring homomorphism
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A function f between rings that has these properties: f(a+b)=f(a)+f(b)
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Image of a function f
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The set of all elements of the codomain to which f maps an element of the domain.
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Extension Field
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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.
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Principal ideal
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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 |
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Left coset
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a+I The set of all elements b of a ring such that a-b is in I.
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Homomorphic image
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S is a homomorphic image of R if f:R→S is a surjective homomorphism of rings
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subring
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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
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True or false finite integral domains are fields
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True`
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t or f Z_p is a field
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true
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Theorem 14.4 R/A is a Field iff A is Maximal
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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.
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The ten axioms to verify Vector Spaces? (u
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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 |
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Subspace
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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
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Span of S
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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)
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Basis of a Vector Space
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S is a basis of V if :
a.) S is linearly independent b.) S spans V |
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Checking if a set is a Basis
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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.
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Coordinate Vector
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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.
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Dimension
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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.
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Transition matrix
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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.
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How to find the transition matrix
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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]
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Cross Product
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|v||u| sin (ø),Only valid in R3 and the cross product is orthogonal to both original vectors.
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Orthogonality
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Two vectors are orthogonal if the result of their dot product is zero.
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Fundamental estimate of linear algebra
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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|
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Steinitz Exchange Theorem
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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
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Exchange Lemma
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Let V be a vector space with E is contained in M a generating set for V containing a linearly independent subset
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Basis
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A linearly independent spanning set
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Dimension
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number of elements in the basis
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Cardinality Criterion of Bases
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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. |
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Dimension Estimate for Vector Subspaces
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If the dimension is the same and its a subspace it isn't a proper subspace.
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Linear Mapping
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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) |
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Rank-Nullity Theorem
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Rank - im (f)
nullity- dim (ker f) f:V->W is linear dim(im f)+dim(ker f) =dim V |
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Layman's steinitz
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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 |
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Cardinality criterion of bases
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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 |
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Cardinality theorems of set
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|L|<|B|<|E| where L is a linearly independent subset B is a Basis E is a generating set
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Dimension estimate for vector subspaces
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A proper subspace of a finitely dimensional vector space has itself a strictly smaller dimension
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Dimension Theorem - Let V be a vector space containing vector subspaces U
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W is in V
dim(u+w)+dim(u n w)=dim U+dimV |
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isomorphism of vector spaces
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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.
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automorphism of a vector space
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an isomorphism of a vector space to itself is called an automorphism of a vector space.
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Fixed point
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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} |
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Complimentary
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two vector subspaces V1,v2 of a vector space V are called complimentary if addition defines a bijection V1 x V2 ~>V
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The classification vector spaces by their dimension
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let n be a natural number. Then a vector space over a field F is isomorphic to F^n iff it has dim n
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Cancellation theorem
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if R is an integral domain and ab=ac then either a=0 or b=c
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Finite integral domain
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a field
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monic
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P in R[x] is monic if its leading coefficient is one
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Division and remainder theorem
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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
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Null ring
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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
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Algebraically closed
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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
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Fundamental Theorem of Algebra
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The field of complex numbers C is algebraically closed
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Determinant
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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)
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cramer's rule
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A. adj A =det (A) (I)
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a matrix is invertible iff
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its determinant is a unit in R
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an endomorphism over a non zero finite dimensional vector space has an eigenvalue if
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it is over an algebraically closed field
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Perron 1907 thm
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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.
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inner product satisfies the following axioms
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(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 |
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a complex inner product satisfies the following axioms
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(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 |
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orthonormal sufficient thm
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let V be a finite dimensional inner product space. Then V has an orthonormal basis.
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norm
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||V||=(v,v)^0.5
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an invertible 3 x 3 matrix that has trace 0
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[ 2 0 0
0 -1 0 0 0 -1] |
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an infinite dimensional vector space
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R[x]
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a vector space with exactly 8 elements
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F^3_2
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a linear mapping f:R^3->R^3 whose rank is 1
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f(x,y,z)=(x,0,0)
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an injective linear mapping from a vector space of dimension 2 to a vector space of dimension 3
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f(x,y)->(x,y,0)
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a non zero idempotent mapping f:V->V that isnt the identity
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abs
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a symmetric bilinear form
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dot product
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an alternating bilinear form
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cross product
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an alternating multi linear form on vx v x v ->R for some non zero real vector space V
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det of 3 real matrix
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an integral domain that is not a field
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R[x]
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a ring that is not commutative
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matrix
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an ideal in a ring that is not principal
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(2,x) in z[x]
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a commutative ring that is not an integral domain
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z/4z
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a non zero polynomial with more roots than its degree
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2(x^2+x) in z/4z[x]
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a commutative ring with unity
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z
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a set with composition
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end(v)
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a field where every element is a unit
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Q
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an algebraically closed field
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complex numbers
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a ring homomorphism
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the inclusion z->Q
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a ring homomorphism f:R->S such that the identity for r isnt the identity of s
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f:R->Mat(2;R) f(x)=[x 0, 0 0] for all x in R
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an ideal of a commutative ring
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mz in z
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principal ideals in a commutative ring
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{0} in R,
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subring of C
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gaussian integers
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diagonalisable nilpotent matrix
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0 matrix
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