Assignment # 2:
Mathematics for Machine Learning
Note: Submit assignment in Hand written in your section class timings. Submission last date:
For Section F4: Wednesday 3rd May and for Section F10: Thursday 4th May.
There will be quiz related to this assignment on same submission date.
Be serious about solving your assignment problems. This assignment will give a practice for solving Mid
Term Exam.
Problem 1: Execute the six steps of worked example 3.3-A Chapter 3 (page 155) to describe the column
space and NULL SPACE and the complete solution of Ax = b.
2 4 6 4
𝐴 = [2 5 7 6]
3 3 5 2
𝑏1
4
𝑏 = [𝑏2 ] = [3]
𝑏3
5
Problem 2: Under what conditions on 𝑏1 , 𝑏2 , 𝑏3 in this system solvable? Include b as fourth column in
elimination. Find all solutions when the conditions hold.
𝑥 + 2𝑦 − 𝑧 = 𝑏1
2𝑥 + 5𝑦 − 4𝑧 = 𝑏2
4𝑥 + 9𝑦 − 8𝑧 = 𝑏3
Problem 3: Consider following matrices. Find out the determinants of each matrix. Then using Echelon
elimination method, obtain the COLUMN SPACE of each matrices. Comparing determinants and echelon
form matrices, decide about independence of vectors. Is there any link of determinant with linearly
independence of vectors.
a) The vectors (1, 3, 2) and (2, 1, 3) and (3, 2, 1)
b) The vectors (1, -3, 2) and (2, 1, -3) and (-3, 2, 1)
Problem 4:
Which number q makes this system singular and which right side t gives it infinitely many solutions? Find
the solution that has z = l.
x + 4y - 2z = l
x + 7y - 6z = 6
3y + qz = t.
Problem 5: Problem 21 of Problem Set 2.2, Problem 29, 30, 31 from Problem set 2.3 from Gilbert Strang
Book, Edition 5
Problem 6: From Problem Set 3.4 (Q: 13, Q: 23) from Gilbert Strang Book, Edition 5
Problem 7: From Problem Set 3.5 (Q: 2, Q: 3) from Gilbert Strang Book, Edition 5
Problem 8: From Problem Set 4.2 (Q: 1, 2, 3, 4,5,6) from Gilbert Strang Book, Edition 5
Problem 9: Related to Orthonormal Vectors and Gram Schmidt process. Before solving the following
problems explain geometrically GRAM-SCHIMDT process for 3-dimensional space.
Find orthonormal basis of following matrices:
1 2 4
a. [0 0 5]
0 3 6
1 −1 2
0
1
0
b. [
]
2
0
0
−1 1 −2
1
1 5
c. [ 2
0 2]
−2 −4 2
Good Luck