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Name: Matriculation number:

September 12, 2019, L.A.G. exam

Solve the following exercises, explaining clearly each passage:

1) Consider the matrix A =

 1 −k

2 − k −1



(1a) Compute the rank of A depending on the parameter k.

(1b) For which values of k is the matrix invertible?

(1c) Compute the inverse of A for the values of k from the previous point 2) Consider the linear transformation f : R

3

→ R

2

defined by

f

 x

1

x

2

x

3

 = x

1

+ x

2

+ x

3

x

2



(2a) Determine the dimension and a basis of Ker(f ) and of Im(f ).

(2b) Say if f is injective and if it is surjective.

(2c) Determine all the vectors v ∈ R

3

, if they exist, such that f (v) = 1 1



3) Consider in the 3-dimensional space the line r of cartesian equation

( x + 2z − 1 = 0 y − z − 1 = 0 (3a) Find a parametric equation of r.

(3b) Determine the plane π perpendicular to r passing through the point (1, 2, 1).

(3c) Find the distance between π and the origin.

(3d) Find the angle between r and the x axis.

4) Consider in the euclidean plane the conic

x

2

+ 2y

2

− 4x − 12y + 21 = 0

(4a) Find a translation X = x + a, Y = y + b that puts the conic in canonical form

(4b) Recognize the type of conic

(4c) Find the coordinates of the centre, if it exists

(4d) Find the coordinates of the vertices.

(2)

Name: Matriculation number:

September 12, 2019, L.A.G. exam

Solve the following exercises, explaining clearly each passage:

1) Consider the matrix A = k k 1 2 − k



(1a) Compute the rank of A depending on the parameter k.

(1b) For which values of k is the matrix invertible?

(1c) Compute the inverse of A for the values of k from the previous point 2) Consider the linear transformation f : R

2

→ R

3

defined by

f x

1

x

2



=

 x

1

+ x

2

x

1

+ x

2

x

1

(2a) Determine the dimension and a basis of Ker(f ) and of Im(f ).

(2b) Say if f is injective and if it is surjective.

(2c) Determine all the vectors v ∈ R

2

, if they exist, such that f (v) =

 1 1 2

3) Consider in the 3-dimensional space the line r of cartesian equation

( x − 2z − 1 = 0 y + z − 1 = 0 (3a) Find a parametric equation of r.

(3b) Determine the cartesian equation of the plane π perpendicular to r passing through the point (1, 1, 1)

(3c) Find the distance between π and the origin.

(3d) Find the angle between r and the x axis.

4) Consider in the euclidean plane the conic x − 2y

2

+ 8y − 9 = 0

(4a) Find a translation X = x + a, Y = y + b that puts the conic in canonical form

(4b) Recognize the type of conic

(4c) Find the coordinates of the centre, if it exists

(4d) Find the coordinates of the vertices.

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