checkanswer.matrix(D, '8313fe0f529090d6a8cdb36248cfdd6c'); # ✅ **<font color=red>Do this:</font>** Find the eigenvalues and eigenvectors of $A$. Set variables ```e1``` and ```vec1``` to be the largest eigenvalue and it's associated eigenvector and ```e2, vec2``` to represent the smallest. # In[ ]: #Put your answer to the above question here. # In[ ]: from answercheck import checkanswer checkanswer.float(e1, "d1bd83a33f1a841ab7fda32449746cc4"); # In[ ]: from answercheck import checkanswer checkanswer.float(e2, "e4c2e8edac362acab7123654b9e73432"); # In[ ]: from answercheck import checkanswer checkanswer.eq_vector(vec1, "09d9df5806bc8ef975074779da1f1023", decimal_accuracy = 4)
# ✅ **<font color=red>QUESTION:</font>** What is a good value to set for ```e``` and why? # Put your answer to the above question here. # ✅ **<font color=red>QUESTION:</font>** The errors seen in this example seem like they would be fairly common in Python. See if you can find a function in ```Numpy``` that has the same purpose as ```checktrue```: # **_Put your answer to the above question here._** # The class ```answercheck``` program will take into consideration round off error. For example, the ```checkanswer.float``` command would consider both of the above correct: # In[7]: from answercheck import checkanswer checkanswer.float(0.300, 'e85b79abfd76b7c13b1334d8d8c194a5') # In[ ]: checkanswer.float(0.1 + 0.2, 'e85b79abfd76b7c13b1334d8d8c194a5') # # # --- # <a name=Solving-Systems-of-Linear-Equations></a> # ## 2. Solving Square Systems of Linear Equations Using Numpy # # # Remember the following set of equations from the mass weight example: # # <img src="https://lh4.googleusercontent.com/48AcnVEBkzXJ_heu4fR0DbP5BBunyRlzPsTeK8WSBsMTSjZ5QNhdhtnVsdAw7wD0NBITIiSmh9Jn0gTPABkxqDa-LrhRicZGdpfbYakgWjJetZfOPk636Eu-vjmj=w740" align="center" width="70%" alt="Image showing two balanced beams, each with three weights. In the top beam is unknown weight A is a distance of 40 to the left of the fulcrum, unknown weight B is a distance of 15 to the left of the fulcrum and a weight of 2 is 50 to the right of the fulcrum. In the bottom beam is the same unknown weights. Weight A is now a distance of 50 to the right of the fulcrum, weight B is a distance of 25 to the left of the fulcrum and the weight of 2 is a distance of 25 to the right of the fulcrum.">
# or:
#
# $$ u \cdot v = \sum^n_{i=1} u_i v_i$$
#
# ✅ **<font color=red>Do This</font>**: Find the dot product of the vectors $u = (1,2,3)$ and $v = (7,9,11)$.
# In[6]:
##Do your work here
# In[7]:
from answercheck import checkanswer
checkanswer.detailedwarnings = False
checkanswer.float(dot, '9ed469ac3b8ef2d21d85e191c8cd24cd')
#
#
# ---
# <a name=Matrix-Multiply></a>
# ## 3. Matrix Multiply
#
# Two matrices $A$ and $B$ can be multiplied together if and only if their "inner dimensions" are the same, i.e. $A$ is
# $m\times d$ and $B$ is $d\times n$ (note that the columns of $A$ and the rows of $B$ are both $d$).
# Multiplication of these two matrices results in a third matrix $C$ with the dimension of $m\times n$.
# Note that $C$ has the same first dimension as $A$ and the same second dimension as $B$. i.e $m\times n$.
#
# _**The $(i,j)$ element in $C$ is the dot product of the $i$th row of $A$ and the $j$th column of $B$.**_
#
# The $i$th row of $A$ is:
def test_float_error():
f = 4.0
with pytest.raises(Exception) as e_info:
checkanswer.float(f, '2e55d74f5c78981f6b877b198bdc61ba')
# \end{matrix}
# \right]
# $$
# In[2]:
#Put your answer here
# In[3]:
from answercheck import checkanswer
checkanswer.float(det,'49afb719e0cd46f74578ebf335290f81');
# ----
#
# <a name="Properties_of_Determinants"></a>
# ## 2. Properties of Determinants
#
# The following are some helpful properties when working with determinants. These properties are often used in proofs and can sometimes be utilized to make faster calculations.
#
# ### Row Operations
#
# Let $A$ be an $n \times n$ matrix and $c$ be a nonzero scalar. Let $|A|$ be a simplified syntax for writing the determinant of $A$:
#
# 1. If a matrix $B$ is obtained from $A$ by multiplying a row (column) by $c$ then $|B| = c|A|$.
# 2. If a matrix $B$ is obtained from $A$ by interchanging two rows (columns) then $|B| = -|A|$.
def test_float_int():
f = 1
checkanswer.float(f, 'e4c2e8edac362acab7123654b9e73432')
def test_float_negative_zero():
f = -0.00
checkanswer.float(f, '30565a8911a6bb487e3745c0ea3c8224')
def test_float_array():
f = [[0.0111]]
checkanswer.float(f, '2e55d74f5c78981f6b877b198bdc61ba')
def test_float_roundoff():
f = 0.011100001
checkanswer.float(f, '2e55d74f5c78981f6b877b198bdc61ba')
def test_float():
f = 0.0111
checkanswer.float(f, '2e55d74f5c78981f6b877b198bdc61ba')
# Put your answer to the above question here
# In[ ]:
##work here
# dist =
# In[ ]:
from answercheck import checkanswer
checkanswer.float(dist,'f8f24a9cedb159fc084bc4e6e347dc07')
# ----
# <a name='innerP'></a>
# ## 6. Inner Products and Matrices
#
# The following is a review from the pre-class assignment.
#
# An inner product on a real vector space $V$ is a function that associates a number, denoted as $\langle u,v \rangle$, with each pair of vectors $u$ and $v$ of $V$. This function satisfies the following conditions for vectors $u, v, w$ and scalar $c$:
#
#
# $$\langle u,v \rangle = \langle v,u \rangle \text{ Symmetry axiom}$$
#
# $$\langle u+v,w \rangle = \langle u,w \rangle + \langle v,w \rangle \text{ Additive axiom}$$
#
# ✅ **<font color=red>QUESTION:</font>** Assuming each team starts with 500,000 fans, what is the steady state of this model? (i.e. in the long term how many Spartan and Wolverine fans will there be?). # In[ ]: #Put your answer here # In[ ]: steadystate # In[ ]: from answercheck import checkanswer checkanswer.float(spartans,'06d263de629f4dbe51eafd524b69ddd9'); # In[ ]: from answercheck import checkanswer checkanswer.float(wolverines,'62d63699c8f7b886ec9b3cb651bba753'); # #
