Linear Algebra

Quiz1 _ Linear Algebra (Python) _ Chapter 1&2&3&4

<h3>&nbsp;</h3>
given the following vector: v=[9 2 74 28 81 4 67 1 9 7 9 128 13], what is the appropriate command code to get the following output: [9, 128, 13] 

<h3>&nbsp;</h3>

&nbsp;
v[:2:]
&nbsp;

&nbsp;
v[::-2]
&nbsp;

&nbsp;
v[-3::]
&nbsp;

&nbsp;
v[::1]
&nbsp;

&nbsp;
v[-1::]
&nbsp;

&nbsp;
choose the correct index of element ‘1’ from the following vector: v=[5 6 6 9 8 0 3 9 7 6 1 5 8]
&nbsp;

&nbsp;
Consider the function &nbsp;&nbsp;R given by f(x) = sin(x1) + ln(x2-x1+1)&nbsp;was an affine function. Find the Taylor approximation &nbsp;f̂ &nbsp;at the point z = (1, 2). Assume that x = (0.96,1.98)
&nbsp;

&nbsp;
What is the length of the given vector notation?&nbsp;Given that a and b are 15-vectors, and c is a 25-vector.&nbsp;
( a,b,c10:21 )&nbsp;
&nbsp;

Using k-means algorithm, group the following points into three clusters: V(97,95,80,80), W(15,70,75,40), Y(20,50,30,20), Z(10,20,70,10), X(50,40,65,45),&nbsp;
Assume random cluster centers. The grouping G results are:

How long would it take to perform a k-means clustering on an n-vector x into K=15 groups?&nbsp;
Where there are 10^3 N vectors with a size of n = 10^6&nbsp;
 
&nbsp;&nbsp;
[Hint: assume the algorithm was run 10 times and the computer can carry 1 Gflop/sec]

&nbsp;&nbsp;15 x 10^16 millisecond&nbsp;

Choose from the following the code that generates a sparse n-vector (where n = 10) with random indices at 1, 5, and 9:

n = 10&nbsp;
w=[1,5,9]&nbsp;
z=np.ones(len(n))&nbsp;
z[w]= np.random.random(len(w))&nbsp;
print(z)&nbsp;

n = 10&nbsp;
w=[1,5,9]&nbsp;
z=np.ones(len(n))&nbsp;
z[w]= np.random.random(len(w))&nbsp;
print(z)&nbsp;
b. n = 10&nbsp;
w = [1, 5, 9]&nbsp;
z=np.sparse(3)&nbsp;
z[w]= np.random(len(w))&nbsp;
print(z)&nbsp;

n = 10&nbsp;
w=[1,5,9]&nbsp;
z=np.sparse(10)&nbsp;
z[w]= np.random.random(len(w))&nbsp;
print(z)&nbsp;

n = 10&nbsp;
w=[1,5,9]&nbsp;
z=np.zeros(10)&nbsp;
z[z]= np.random.random(len(w))&nbsp;
print(z)&nbsp;

n=10&nbsp;
w=[1,5,9]&nbsp;
z=np.zeros(10)&nbsp;
z[w]= np.random.random(len(w))&nbsp;
print(z) &nbsp;

&nbsp;Which of the following means the vectors v and u have the same length?&nbsp;

‖ u − v ‖ = ‖ u ‖ + ‖ v ‖

|u . u| = |v . v|

‖ u + v ‖ = ‖ u ‖ + ‖ v ‖

u / ‖ u ‖ = v / ‖ v ‖

‖ u − v ‖ = ‖ u ‖ - ‖ v ‖

&nbsp;The distance between ℎ1 = [2.2 4.8 1.1 2.1 4.1] and ℎ2 = [4.8 0.7 0.1 1.1 0.6] is: &nbsp;

What is the angle (in degrees) between the vectors?&nbsp;
u = [1,4,6] and &nbsp;v = [1,2,3].