Coding the Matrix Week 1 The Vector Space作业

 本周的作业较少，只有一个编程任务hw2.作业比较简单，如果大学学习过矩阵代数的话，基本上没有什么问题，不过要注意的一点是基2的Span的求法。

  基2空间上，在所有基向量中取任意个数个，叠加组合就得到了Span。但是如何取任意个呢？下面给出几种方法。

  一种方法是对于任意可能的个数，利用Python中的排列组合module生成对应于此个数的所有排列，即得到Span。感兴趣的话可以百度一下。这种方法概念较为清晰，但需要对Python的库较为了解。

  第二种方法，利用了从n个元素中任选任意个的方法只有2^n个的排列组合知识。具体来说，就是对于任意从0到2^n-1的整数，使用bin()函数得到其二进制表示，对应位为1即代表选中此基向量。这种方法稍微Smart一些。但过程较为冗杂。

  第三种，就是下面程序给出的方法，一行语句就可以完成运算，原理和上一种方法相同，就不再赘述。但要注意结果为0的情况下要单独考虑。

  其他的向量运算都比较简单，最后几道判断是否为向量空间的问题，只需要牢记向量空间三特征（包含0，加减和向量乘法组合仍在空间内）就不会出错。

  作业代码如下，模版中注释部分给出了验证范例。

# version code 761   
# Please fill out this stencil and submit using the provided submission script.   
  
from vec import Vec  
from GF2 import one  
  
  
## Problem 1   
def vec_select(veclist, k):   
    ''''' 
    >>> D = {'a','b','c'} 
    >>> v1 = Vec(D, {'a': 1}) 
    >>> v2 = Vec(D, {'a': 0, 'b': 1}) 
    >>> v3 = Vec(D, {        'b': 2}) 
    >>> v4 = Vec(D, {'a': 10, 'b': 10}) 
    >>> vec_select([v1, v2, v3, v4], 'a') == [Vec(D,{'b': 1}), Vec(D,{'b': 2})] 
    True 
    '''  
    return [x for x in veclist if x[k]==0]  
  
def vec_sum(veclist, D):   
    ''''' 
    >>> D = {'a','b','c'} 
    >>> v1 = Vec(D, {'a': 1}) 
    >>> v2 = Vec(D, {'a': 0, 'b': 1}) 
    >>> v3 = Vec(D, {        'b': 2}) 
    >>> v4 = Vec(D, {'a': 10, 'b': 10}) 
    >>> vec_sum([v1, v2, v3, v4], D) == Vec(D, {'b': 13, 'a': 11}) 
    True 
    '''  
    return sum(veclist) if len(veclist)!=0 else Vec(D,{})  
  
def vec_select_sum(veclist, k, D):   
    ''''' 
    >>> D = {'a','b','c'} 
    >>> v1 = Vec(D, {'a': 1}) 
    >>> v2 = Vec(D, {'a': 0, 'b': 1}) 
    >>> v3 = Vec(D, {        'b': 2}) 
    >>> v4 = Vec(D, {'a': 10, 'b': 10}) 
    >>> vec_select_sum([v1, v2, v3, v4], 'a', D) == Vec(D, {'b': 3}) 
    True 
    '''  
    return vec_sum(vec_select(veclist,k),D)  
  
  
  
## Problem 2   
def scale_vecs(vecdict):  
    ''''' 
    >>> v1 = Vec({1,2,3}, {2: 9}) 
    >>> v2 = Vec({1,2,4}, {1: 1, 2: 2, 4: 8}) 
    >>> scale_vecs({3: v1, 5: v2}) == [Vec({1,2,3},{2: 3.0}), Vec({1,2,4},{1: 0.2, 2: 0.4, 4: 1.6})] 
    True 
    '''  
    return [y/x for (x,y) in vecdict.items()]  
  
  
  
## Problem 3   
def GF2_span(D, L):  
    ''''' 
    >>> from GF2 import one 
    >>> D = {'a', 'b', 'c'} 
    >>> L = [Vec(D, {'a': one, 'c': one}), Vec(D, {'b': one})] 
    >>> len(GF2_span(D, L)) 
    4 
    >>> Vec(D, {}) in GF2_span(D, L) 
    True 
    >>> Vec(D, {'b': one}) in GF2_span(D, L) 
    True 
    >>> Vec(D, {'a':one, 'c':one}) in GF2_span(D, L) 
    True 
    >>> Vec(D, {x:one for x in D}) in GF2_span(D, L) 
    True 
    '''  
    if len(L)==0:return []  
    maxind=2**len(L)-1  
    res=[sum([L[j] for j in range(len(L)) if i//(2**j)%2]) for i in range(maxind+1)]  
    res.append(Vec(D,{}))  
    del res[0]  
    return res  
  
  
  
## Problem 4   
# Answer with a boolean, please.   
  
is_it_a_vector_space_1 = True  
is_it_a_vector_space_2 = False  
  
  
  
## Problem 5   
is_it_a_vector_space_3 = True  
is_it_a_vector_space_4 = False  
  
  
## Problem 6   
  
is_it_a_vector_space_5 = True  
is_it_a_vector_space_6 = False  

# version code 761
# Please fill out this stencil and submit using the provided submission script.

from vec import Vec
from GF2 import one


## Problem 1
def vec_select(veclist, k): 
    '''
    >>> D = {'a','b','c'}
    >>> v1 = Vec(D, {'a': 1})
    >>> v2 = Vec(D, {'a': 0, 'b': 1})
    >>> v3 = Vec(D, {        'b': 2})
    >>> v4 = Vec(D, {'a': 10, 'b': 10})
    >>> vec_select([v1, v2, v3, v4], 'a') == [Vec(D,{'b': 1}), Vec(D,{'b': 2})]
    True
    '''
    return [x for x in veclist if x[k]==0]

def vec_sum(veclist, D): 
    '''
    >>> D = {'a','b','c'}
    >>> v1 = Vec(D, {'a': 1})
    >>> v2 = Vec(D, {'a': 0, 'b': 1})
    >>> v3 = Vec(D, {        'b': 2})
    >>> v4 = Vec(D, {'a': 10, 'b': 10})
    >>> vec_sum([v1, v2, v3, v4], D) == Vec(D, {'b': 13, 'a': 11})
    True
    '''
    return sum(veclist) if len(veclist)!=0 else Vec(D,{})

def vec_select_sum(veclist, k, D): 
    '''
    >>> D = {'a','b','c'}
    >>> v1 = Vec(D, {'a': 1})
    >>> v2 = Vec(D, {'a': 0, 'b': 1})
    >>> v3 = Vec(D, {        'b': 2})
    >>> v4 = Vec(D, {'a': 10, 'b': 10})
    >>> vec_select_sum([v1, v2, v3, v4], 'a', D) == Vec(D, {'b': 3})
    True
    '''
    return vec_sum(vec_select(veclist,k),D)



## Problem 2
def scale_vecs(vecdict):
    '''
    >>> v1 = Vec({1,2,3}, {2: 9})
    >>> v2 = Vec({1,2,4}, {1: 1, 2: 2, 4: 8})
    >>> scale_vecs({3: v1, 5: v2}) == [Vec({1,2,3},{2: 3.0}), Vec({1,2,4},{1: 0.2, 2: 0.4, 4: 1.6})]
    True
    '''
    return [y/x for (x,y) in vecdict.items()]



## Problem 3
def GF2_span(D, L):
    '''
    >>> from GF2 import one
    >>> D = {'a', 'b', 'c'}
    >>> L = [Vec(D, {'a': one, 'c': one}), Vec(D, {'b': one})]
    >>> len(GF2_span(D, L))
    4
    >>> Vec(D, {}) in GF2_span(D, L)
    True
    >>> Vec(D, {'b': one}) in GF2_span(D, L)
    True
    >>> Vec(D, {'a':one, 'c':one}) in GF2_span(D, L)
    True
    >>> Vec(D, {x:one for x in D}) in GF2_span(D, L)
    True
    '''
    if len(L)==0:return []
    maxind=2**len(L)-1
    res=[sum([L[j] for j in range(len(L)) if i//(2**j)%2]) for i in range(maxind+1)]
    res.append(Vec(D,{}))
    del res[0]
    return res



## Problem 4
# Answer with a boolean, please.

is_it_a_vector_space_1 = True
is_it_a_vector_space_2 = False



## Problem 5
is_it_a_vector_space_3 = True
is_it_a_vector_space_4 = False


## Problem 6

is_it_a_vector_space_5 = True
is_it_a_vector_space_6 = False

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