hw5.py

# version code 941
# Please fill out this stencil and submit using the provided submission script.

from vecutil import list2vec
from solver import solve
from matutil import listlist2mat, coldict2mat, mat2rowdict, mat2coldict
from mat import Mat
from GF2 import one
from vec import Vec
from hw4 import vec2rep, rep2vec
from hw4 import is_superfluous
from independence import rank
from triangular import triangular_solve_n


## Problem 1
w0 = list2vec([1,0,0])
w1 = list2vec([0,1,0])
w2 = list2vec([0,0,1])

v0 = list2vec([1,2,3])
v1 = list2vec([1,3,3])
v2 = list2vec([0,3,3])

# Fill in exchange_S1 and exchange_S2
# with appropriate lists of 3 vectors

exchange_S0 = [w0, w1, w2]
exchange_S1 = [v0, w1, w2]
exchange_S2 = [v0, v1, w2]
exchange_S3 = [v0, v1, v2]


## Problem 2
w0 = list2vec([0,one,0])
w1 = list2vec([0,0,one])
w2 = list2vec([one,one,one])

v0 = list2vec([one,0,one])
v1 = list2vec([one,0,0])
v2 = list2vec([one,one,0])

exchange_2_S0 = [w0, w1, w2]
exchange_2_S1 = [v0, w1, w2]
exchange_2_S2 = [v0, v1, w2]
exchange_2_S3 = [v0, v1, v2]


## Problem 3
def morph(S, B):
    '''
    Input:
        - S: a list of distinct Vec instances
        - B: a list of linearly independent Vec instances
        - Span S == Span B
    Output: a list of pairs of vectors to inject and eject
    Example:
        >>> #This is how our morph works.  Yours may yield different results.
        >>> S = [list2vec(v) for v in [[1,0,0],[0,1,0],[0,0,1]]]
        >>> B = [list2vec(v) for v in [[1,1,0],[0,1,1],[1,0,1]]]
        >>> morph(S, B)
        [(Vec({0, 1, 2},{0: 1, 1: 1, 2: 0}), Vec({0, 1, 2},{0: 1, 1: 0, 2: 0})), (Vec({0, 1, 2},{0: 0, 1: 1, 2: 1}), Vec({0, 1, 2},{0: 0, 1: 1, 2: 0})), (Vec({0, 1, 2},{0: 1, 1: 0, 2: 1}), Vec({0, 1, 2},{0: 0, 1: 0, 2: 1}))]

    '''
    
    pairs = []                                                # Initialize to empty set of (inject, eject) pairs
    scpy = S[:]
    for inject in B:                                          # Inject one vector at a time from B into S.
        scpy = scpy + [inject]
        for i in range(len(scpy)):                            # Take one vector at a time in bigger set
            if scpy[i] == inject: continue                    # We don no want to test the vector we injected
            u = vec2rep (scpy[:i]+scpy[i+1:], scpy[i])        # See if it can be written as a linear combination of the remaining vectors
            if u is not None:                                 # If it can be expressed a linear combination , then it can be ejected 
                pairs.append((inject, scpy[i]))
                del(scpy[i])
                break                                         # Move on to the next vector to inject
    return pairs


## Problem 4
# Please express each solution as a list of vectors (Vec instances)

row_space_1 = [list2vec(v) for v in [[1, 2, 0], [0, 0, 1]]]
col_space_1 = [list2vec(v) for v in [[1, 0], [0, 1]]]

row_space_2 = [list2vec(v) for v in [[1, 4, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]]
col_space_2 = [list2vec(v) for v in [[1, 0, 0], [0, 1, 0], [0, 0, 1]]]

row_space_3 = [list2vec(v) for v in [[1]]]
col_space_3 = [list2vec(v) for v in [[1, 2, 3]]]

row_space_4 = [list2vec(v) for v in [[1, 0], [0, 1]]]
col_space_4 = [list2vec(v) for v in [[1, 2, 3], [0, 0, 5]]]


## Problem 5
def my_is_independent(L): 
    '''
    input:  A list, L, of Vecs
    output: A boolean indicating if the list is linearly independent
    
    >>> L = [Vec({0, 1, 2},{0: 1, 1: 0, 2: 0}), Vec({0, 1, 2},{0: 0, 1: 1, 2: 0}), Vec({0, 1, 2},{0: 0, 1: 0, 2: 1}), Vec({0, 1, 2},{0: 1, 1: 1, 2: 1}), Vec({0, 1, 2},{0: 1, 1: 1, 2: 0}), Vec({0, 1, 2},{0: 0, 1: 1, 2: 1})]
    >>> my_is_independent(L)
    False
    >>> my_is_independent(L[:2])
    True
    >>> my_is_independent(L[:3])
    True
    >>> my_is_independent(L[1:4])
    True
    >>> my_is_independent(L[0:4])
    False
    >>> my_is_independent(L[2:])
    False
    >>> my_is_independent(L[2:5])
    False
    '''
    if rank(L) == len(L): return True
    else: return False


## Problem 6
def subset_basis(T): 
    '''
    input: A list, T, of Vecs
    output: A list, S, containing Vecs from T, that is a basis for the
    space spanned by T.
    
    >>> a0 = Vec({'a','b','c','d'}, {'a':1})
    >>> a1 = Vec({'a','b','c','d'}, {'b':1})
    >>> a2 = Vec({'a','b','c','d'}, {'c':1})
    >>> a3 = Vec({'a','b','c','d'}, {'a':1,'c':3})
    >>> subset_basis([a0,a1,a2,a3]) == [Vec({'c', 'b', 'a', 'd'},{'a': 1}), Vec({'c', 'b', 'a', 'd'},{'b': 1}), Vec({'c', 'b', 'a', 'd'},{'c': 1})]
    True
    '''

    # Subset basis is not unique.  Depends on where one starts.
    # For example the above problems also has a basis given by
    #    >>> subset_basis([a0,a1,a2,a3]) == [Vec({'c', 'b', 'a', 'd'},{'b': 1}), Vec({'c', 'b', 'a', 'd'},{'c': 1}), Vec({'c', 'b', 'a', 'd'},{'a':1, 'c': 3})]

    inT = T[:]
    for i in range(len(list(inT))):
        if is_superfluous (inT, i): 
            del(inT[i])
            break

    return inT

## Problem 7
def my_rank(L): 
    '''
    input: A list, L, of Vecs
    output: The rank of the list of Vecs
    
    >>> my_rank([list2vec(v) for v in [[1,2,3],[4,5,6],[1.1,1.1,1.1]]])
    2
    '''
    return (rank(subset_basis(L)))


## Problem 8
# Please give each answer as a boolean

only_share_the_zero_vector_1 = True # There is a 1 in non overlapping positions in the vector from U and V
only_share_the_zero_vector_2 = True
only_share_the_zero_vector_3 = True


## Problem 9
def direct_sum_decompose(U_basis, V_basis, w):
    '''
    input:  A list of Vecs, U_basis, containing a basis for a vector space, U.
    A list of Vecs, V_basis, containing a basis for a vector space, V.
    A Vec, w, that belongs to the direct sum of these spaces.
    output: A pair, (u, v), such that u+v=w and u is an element of U and
    v is an element of V.
    
    >>> U_basis = [Vec({0, 1, 2, 3, 4, 5},{0: 2, 1: 1, 2: 0, 3: 0, 4: 6, 5: 0}), Vec({0, 1, 2, 3, 4, 5},{0: 11, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0}), Vec({0, 1, 2, 3, 4, 5},{0: 3, 1: 1.5, 2: 0, 3: 0, 4: 7.5, 5: 0})]
    >>> V_basis = [Vec({0, 1, 2, 3, 4, 5},{0: 0, 1: 0, 2: 7, 3: 0, 4: 0, 5: 1}), Vec({0, 1, 2, 3, 4, 5},{0: 0, 1: 0, 2: 15, 3: 0, 4: 0, 5: 2})]
    >>> w = Vec({0, 1, 2, 3, 4, 5},{0: 2, 1: 5, 2: 0, 3: 0, 4: 1, 5: 0})
    >>> direct_sum_decompose(U_basis, V_basis, w) == (Vec({0, 1, 2, 3, 4, 5},{0: 2.0, 1: 4.999999999999972, 2: 0.0, 3: 0.0, 4: 1.0, 5: 0.0}), Vec({0, 1, 2, 3, 4, 5},{0: 0.0, 1: 0.0, 2: 0.0, 3: 0.0, 4: 0.0, 5: 0.0}))
    True
    '''
    dsum_basis = U_basis + V_basis                              # Basis of the direct sum is the union of the bases of the sub spaces
    sol = vec2rep (dsum_basis, w)                               # get the linear solution that multiplies with dsum_basis that gives w
    com = list (sol.f.values())                                 # break the U and V parts of the solution
    uvals = list2vec (com[:len(U_basis)])                            
    vvals = list2vec (com[len(U_basis):])     
    u = rep2vec(uvals, U_basis)
    v = rep2vec(vvals, V_basis)
    return u, v                       
    
## Problem 10
def is_invertible(M): 
    '''
    input: A matrix, M
    output: A boolean indicating if M is invertible.
    
    >>> M = Mat(({0, 1, 2, 3}, {0, 1, 2, 3}), {(0, 1): 0, (1, 2): 1, (3, 2): 0, (0, 0): 1, (3, 3): 4, (3, 0): 0, (3, 1): 0, (1, 1): 2, (2, 1): 0, (0, 2): 1, (2, 0): 0, (1, 3): 0, (2, 3): 1, (2, 2): 3, (1, 0): 0, (0, 3): 0})
    >>> is_invertible(M)
    True
    '''
    
    if M.D[0] != M.D[1]: return False
    
    rowdict = mat2rowdict(M)
    rowlist = list (rowdict.values())
    coldict = mat2coldict (M)
    collist = list(coldict.values())
    # check if the matrix is square and the columns are independent
    if rank(rowlist) == rank (collist) and len (collist) == rank (collist): return True
    else: return False
    

## Problem 11
def find_matrix_inverse(A):
    '''
    input: An invertible matrix, A, over GF(2)
    output: Inverse of A

    >>> M = Mat(({0, 1, 2}, {0, 1, 2}), {(0, 1): one, (1, 2): 0, (0, 0): 0, (2, 0): 0, (1, 0): one, (2, 2): one, (0, 2): 0, (2, 1): 0, (1, 1): 0})
    >>> find_matrix_inverse(M) == Mat(({0, 1, 2}, {0, 1, 2}), {(0, 1): one, (2, 0): 0, (0, 0): 0, (2, 2): one, (1, 0): one, (1, 2): 0, (1, 1): 0, (2, 1): 0, (0, 2): 0})
    True
    '''
    
    B = {}
    for i in A.D[1]:
        v = Vec(A.D[0], {i:one})
        u = solve (A, v)
        B[i] = u
    
    return coldict2mat(B)

## Problem 12
def find_triangular_matrix_inverse(A): 
    '''
    input: An upper triangular Mat, A, with nonzero diagonal elements
    output: Inverse of A
    >>> A = listlist2mat([[1, .5, .2, 4],[0, 1, .3, .9],[0,0,1,.1],[0,0,0,1]])
    >>> find_triangular_matrix_inverse(A) == Mat(({0, 1, 2, 3}, {0, 1, 2, 3}), {(0, 1): -0.5, (1, 2): -0.3, (3, 2): 0.0, (0, 0): 1.0, (3, 3): 1.0, (3, 0): 0.0, (3, 1): 0.0, (2, 1): 0.0, (0, 2): -0.05000000000000002, (2, 0): 0.0, (1, 3): -0.87, (2, 3): -0.1, (2, 2): 1.0, (1, 0): 0.0, (0, 3): -3.545, (1, 1): 1.0})
    True
    '''
    B = {}
    rows = mat2rowdict(A)
    for i in A.D[1]:
        v = Vec(A.D[0], {i:1})
        u = triangular_solve_n (rows, v)
        B[i] = u
    
    return coldict2mat(B)