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TRENTO, A.A. 2019/20

MATHEMATICS FOR DATA SCIENCE/BIOSTATISTICS EXERCISE SHEET # 5

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.

Important! Reduce all of the complete matrices in thwe following exercises to RREF forms.

Exercise 5.1. State whether the system in the unknowns x

1

, x

2

has solutions, and in case find them all. {

x

1

+ x

2

= 2 2x

1

+ 2x

2

= 3

Exercise 5.2. State whether the system in the unknowns x

1

, x

2

has solutions, and in case find them all. {

x

1

+ x

2

= 2 2x

1

+ 2x

2

= 4

Exercise 5.3. State whether the system in the unknowns x

1

, x

2

, x

3

has solutions, and in case find them all. {

x

1

+ x

2

= 2 x

1

+ 2x

2

= 3

Exercise 5.4. State whether the system in the unknowns x

1

, x

2

, x

3

has solutions, and in case find them all. 

 

 

x

1

+ x

2

+ x

3

= 0 x

1

+ 2x

2

+ 3x

3

= 0 x

1

− x

2

− x

3

= 0

Exercise 5.5. State whether the system in the unknowns x

1

, x

2

, x

3

has solutions, and in case find them all. 

 

 

x

1

+ x

2

+ x

3

= 0 x

1

+ 2x

2

+ 3x

3

= 0 x

1

− x

2

− 3x

3

= 0

Exercise 5.6. State whether the system in the unknowns x

1

, x

2

, x

3

has solutions, and in case find them all. 

 

 

x

1

+ x

2

+ x

3

= 1

x

1

+ 2x

2

+ 3x

3

= 2

x

1

− x

2

− x

3

= 3

(2)

2 EXERCISE SHEET # 5

Exercise 5.7. State whether the system in the unknowns x

1

, x

2

, x

3

has solutions, and in case find them all.

 

 

x

1

+ x

2

+ x

3

= 1 x

1

+ 2x

2

+ 3x

3

= 2 x

1

− x

2

− 3x

3

= 3

Exercise 5.8. State whether the system in the unknowns x

1

, x

2

, x

3

, x

4

has solutions, and in case find them all.

 

 

x

1

+ x

2

+ x

3

+ x

4

= 1 x

1

− 2x

2

− x

3

− 3x

4

= 1 3x

1

− 3x

2

− x

3

− 5x

4

= 0.

Exercise 5.9. State whether the system in the unknowns x

1

, x

2

, x

3

, x

4

has solutions, and in case find them all.

 

 

x

1

+ x

2

+ x

3

+ x

4

= 1 x

1

− 2x

2

− x

3

− 3x

4

= 1 3x

1

− 3x

2

− x

3

− 5x

4

= 3.

Exercise 5.10.

(1) Define

(a) systems of generators, (b) linear dependence,

(c) linear independence, (d) bases.

(2) Show that the following are equivalent.

(a) v

1

, . . . , v

m

are linearly dependent, and

(b) (at least) one of the v

i

can be written as a linear combination of the others.

Show that as a consequence, if v

1

, . . . , v

m

are a system of generators of a space V , and they are linearly dependent, say v

m

is a linear combination of v

1

, . . . , v

m−1

, then v

1

, . . . , v

m−1

is a system of generators for V , that is, you can safely drop v

m

.

(3) Show that the following are equivalent.

(a) v

1

, . . . , v

m

are a basis of the space V , and

(b) Every element of V can be written uniquely as a linear combination of v

1

, . . . , v

m

.

Exercise 5.11. Show that if one of v

1

, . . . , v

m

is zero, then v

1

, . . . , v

m

are linearly dependent.

Exercise 5.12. For each of the following sysyem of vectors, state whether they are

a basis of R

3

. When they are not, state explicitly which condition fails.

(3)

EXERCISE SHEET # 5 3

(1) 

 

 

v

1

= (1, 0, 0) v

2

= (0, 1, 0) v

3

= (0, 0, 1)

(2) {

v

1

= (1, 0, 0) v

2

= (0, 1, 0)

(3) 

 

 

v

1

= (1, −1, 0) v

2

= (0, 1, −1) v

3

= ( −1, 0, 1)

(4) 

 

 

v

1

= (1, −1, 0)

v

2

= (0, 1, −1)

v

3

= (1, 0, 1)

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