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Math Help - [SOLVED] URGENT exam help needed!!

  1. #1
    iarapovic
    Guest

    [SOLVED] URGENT exam help needed!!

    Hey Everyone,
    Tomorro i have an exam Linear Algebra, and i have been given a mock exam to practice on. Unfortunatrly my lecturer decided not to print the solutions to the mock exam and i have no way if knowing what i am doing is right!! Can i please have some answers?

    1.Define the terms eigenspace, generalised eigenspace for an operator on Rn.

    2.Show that the dimension of the eigenspace cannot be greater than the dimension of the generalised eigenspace.

    3. Define the terms nilpotent matrix, nilpotent operator
    on a vector space.

    4.By finding eigenvalues of the matrix
    A = [4 -1]
    [1 2]
    show A is the sum of a nilpotent and a diagonal matrix, and exponentiate A. Using this result, write down the solution of the system of Ordinary
    Differential Equations x' = Ax

    5. Define the terms ring, field, polynomial of degree n over a field F, root of a polynomial, polynomial function, and finally zero of a polynomial function.

    6.Define the terms homomorphism of groups,
    isomorphism of groups. Define groups and subgroups.

    7.Define the term onto, one-one and bijective for maps from a set A to a set B. Show that the composite of onto maps is onto!
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  2. #2
    MHF Contributor kalagota's Avatar
    Joined
    Oct 2007
    From
    Taguig City, Philippines
    Posts
    1,026
    Quote Originally Posted by iarapovic View Post
    Hey Everyone,
    Tomorro i have an exam Linear Algebra, and i have been given a mock exam to practice on. Unfortunatrly my lecturer decided not to print the solutions to the mock exam and i have no way if knowing what i am doing is right!! Can i please have some answers?

    1.Define the terms eigenspace, generalised eigenspace for an operator on Rn.

    2.Show that the dimension of the eigenspace cannot be greater than the dimension of the generalised eigenspace.

    3. Define the terms nilpotent matrix, nilpotent operator
    on a vector space.

    4.By finding eigenvalues of the matrix
    A = [4 -1]
    [1 2]
    show A is the sum of a nilpotent and a diagonal matrix, and exponentiate A. Using this result, write down the solution of the system of Ordinary
    Differential Equations x' = Ax

    5. Define the terms ring, field, polynomial of degree n over a field F, root of a polynomial, polynomial function, and finally zero of a polynomial function.

    6.Define the terms homomorphism of groups,
    isomorphism of groups. Define groups and subgroups.

    7.Define the term onto, one-one and bijective for maps from a set A to a set B. Show that the composite of onto maps is onto!

    okay, i'll try to answer some (and i'll answer it from 7-1, in accordance to the sequence of our study)..

    Let f: A \rightarrow B be a function (to be precise).
    f is said to be one-to-one if f(x)=f(y) \implies x=y
    f is said to be onto B if the range of f is B.
    if f is onto and one-to-one, then f is a bijection..

    A (nonempty) set G with a binary operation, say *, is said to be a group if the following are satisfied:
    i) * is associative, i.e. a*(b*c)=(a*b)*c
    ii) there exists an element e in G such that for every elements g in G, e*g=g=g*e (e is called the identity element of G)
    iii) for every elements g in G, there exists an inverse g^{-1} in G such that g*g^{-1} = e = g^{-1}*g

    note: if * is commutative, i.e. for all a,b in G, a*b=b*a, then G is said to be an abelian group.

    Let H be a (nonempty) subset of G. H is said to be a subgroup of G iff H is a group under the operation of G.

    Let \phi be a map of a group G into a group G'.

    \phi is a homomorphism if for all a,b in G \phi (ab) = \phi (a) \phi (b)..(here we note that on the LHS, the operation of G is used; while on the RHS, the operation of G' is used.. also, both operation may not be distinct..)

    A group homomorphism \phi : G \rightarrow G' is said to be an isomorphism if \phi is a bijection..

    A (nonempty) set R with two binary operations, + and \cdot, is said to be a ring if the following are satisfied:
    i) <R,+> is an abelian group. (R is an abelian group with respect to +)
    ii) \cdot is associative
    iii) for all a,b,c in R, a \cdot (b+c) = a \cdot b + a \cdot c (Left Distributive Law);  (a+b) \cdot c = a \cdot c + b \cdot c (Right Distributive Law)

    A (nonempty) set F with two binary operations, + and \cdot, is said to be a field if the following are satisfied:
    i) <F,+, \cdot> is a ring.
    ii) \cdot is commutative
    iii) there exists an element 1_F in F such that for all elements a in F, 1_F \cdot a = a = a \cdot 1_F ( 1_F (or 1) is called the multiplicative identity element of F)
    iv) for every element a in F s.t a \neq 0, there exists an element a^{-1} in F st a \cdot a^{-1} = 1_F = a^{-1} \cdot a ( a^{-1} is called the multiplicative inverse of a) (also note that, this time, -a denotes the additive inverse of a; and 0 is the additive identity in F)
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