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TRENTO, A.A. 2019/20

MATHEMATICS FOR DATA SCIENCE/BIOSTATISTICS EXERCISE SHEET # 10

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.

Notice! For technical reasons, sometimes in the following exercises we write something like “e

1

+ ( −1)e

2

”, which just means e

1

− e

2

.

Exercise 10.1. Let M

2×2

be the vector space of 2 × 2 real matrices.

We recall that dim(V ) = 4.

• Prove that the matrices A =

[ 0 1 2 0

]

, B =

[ 1 2 3 0

]

are linearly independent.

• Extend A, B to a basis of M

2×2

.

Exercise 10.2. Let M

2×3

be the vector space of 2 × 3 real matrices.

We recall that dim(V ) = 6.

• Prove that the matrices A =

[ 0 1 2 2 0 0

]

, B =

[ 1 2 4 3 0 1

]

, C =

[ 1 0 1 0 1 1

]

are linearly independent.

• Extend A, B, C to a basis of M

2×3

.

Exercise 10.3. Let M

2×2

be the vector space of 2 × 2 real matrices.

Let V = span {A, B, C, D}, where

A =

[ 0 1 2 0

]

, B =

[ 1 2 3 0

]

, C =

[ 1 2 3 4

]

, D =

[ 1 2 3 1

] . Extract from A, B, C, D a basis of V .

Is V = M

2×2

?

Exercise 10.4. Let V be a vector space of dimension 2, and let e

1

, e

2

be a base of V .

Consider the linear function f : V → V given by {

f (e

1

) = e

1

+ e

2

f (e

2

) = ( −1)e

1

+ 2e

2

Write the matrix A of f with respect to the base e

1

, e

2

.

(2)

2 EXERCISE SHEET # 10

Consider the vectors {

g

1

= e

1

+ ( −1)e

2

, g

2

= 2e

1

+ e

2

. (1) Show that g

1

, g

2

are a base of V ,

(2) write e

1

, e

2

as linear combinations of g

1

, g

2

, and

(3) write the matrix B of f with respect to the base g

1

, g

2

.

Exercise 10.5. Let V be a vector space of dimension 2, and let e

1

, e

2

be a base of V .

Consider the linear function f : V → V given by {

f (e

1

) = 5e

1

+ 2e

2

f (e

2

) = ( −12)e

1

+ ( −5)e

2

Write the matrix A of f with respect to the base e

1

, e

2

. Consider the vectors {

g

1

= 3e

1

+ e

2

, g

2

= 2e

1

+ e

2

. (1) Show that g

1

, g

2

are a base of V ,

(2) write e

1

, e

2

as linear combinations of g

1

, g

2

, and

(3) write the matrix B of f with respect to the base g

1

, g

2

.

Exercise 10.6. Let V be a vector space of dimension 2, and let e

1

, e

2

be a base of V .

Consider the linear function f : V → V given by {

f (e

1

) = ( −4)e

1

+ 10e

2

f (e

2

) = ( −3)e

1

+ 7e

2

Write the matrix A of f with respect to the base e

1

, e

2

.

Consider the vectors {

g

1

= 3e

1

+ ( −5)e

2

, g

2

= ( −1)e

1

+ 2e

2

. (1) Show that g

1

, g

2

are a base of V ,

(2) write e

1

, e

2

as linear combinations of g

1

, g

2

, and

(3) write the matrix B of f with respect to the base g

1

, g

2

.

Exercise 10.7. Use the rules of Laplace and Sarrus to compute some of the deter- minants of sheet #8.

Exercise 10.8. Let V be a vector space of dimension 3, and let e

1

, e

2

, e

3

be a base of V .

Consider the linear function f : V → V given by f (e

1

) = e

1

+ e

2

f (e

2

) = e

2

+ e

3

f (e

3

) = e

1

− e

3

(3)

EXERCISE SHEET # 10 3

Write the matrix A of f with respect to the base e

1

, e

2

, e

3

. Consider the vectors

g

1

= e

1

,

g

2

= e

1

− e

2

− e

3

,

g

3

= e

3

.

(1) Show that g

1

, g

2

, g

3

are a base of V ,

(2) write e

1

, e

2

, e

3

as linear combinations of g

1

, g

2

, g

3

, and

(3) write the matrix B of f with respect to the base g

1

, g

2

, g

3

.

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