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TRENTO, A.A. 2019/20

MATHEMATICS FOR DATA SCIENCE/BIOSTATISTICS EXERCISE SHEET # 3

Important! In solving the exercises

• explain what you are doing,

• explain why you are doing what you are doing, and

• spell out all intermediate steps.

Exercise 3.1. Consider the matrix

A =

 

1 −1 1 2

2 0 −1 −1

0 1 0 1

1 1 −1 −1

 

 .

Compute

S

13

A, S

24

A, E

14

( −1)A, E

21

( −2)A.D

2

(1/2)A, D

4

( −1)A.

Recall that the elementary row operations are

(1) Exchanging two rows. (We will denote by S

ij

the operation which exhanges the i-th row and the j-th one.)

(2) Adding to a row a scalar multiple of another. (We will denote by E

ij

(c) the operation that replaces the i-th row with the sum of the i-th row and the multiple by the scalar c of the j-th row.)

(3) Multiplying a row by a non-zero scalar. (We will denote by D

i

(c) the operation that multiplies the i-th row by the scalar c ̸= 0.)

Exercise 3.2. Find the solutions of the following systems in x, y, z, by putting in REF form the associated matrices, and try and give a geometric interpretation.

{

x + y + z = 0 x − y + 2z = 0

 

 

x + y + z = 0 x − y + 2z = 0 3x − y + 5z = 0

 

 

x + y + z = 0

x − y + 2z = 0

3x − y + 4z = 0

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