**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 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
### 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}