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1) Consider the matrix M = kI + A · A

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Name: Matriculation number:

February 6, 2020, L.A.G. exam

Solve the following exercises, explaining clearly each passage:

1) Consider the matrix M = kI + A · A

where k is a real parameter, I is the 2 × 2 identity matrix and A = 1 1 1 −1



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

(b) For which values of k is M invertible?

(c) Compute the inverse of M for the values of k in (b).

2) Consider the linear transformation f : R

3

→ R

2

defined by

f (

 x y z

) = x 1 1

 + y  1

−1



− z 1 1



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

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

(c) Determine all the vectors v ∈ R

3

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



3) In the 3-dim. euclidean space consider the line r passing through the origin, and perpendicular to the plane x − z = 400, and the line s passing through the point (1, 0, 1) and parallel to the vector (0, 500, −500).

(a) Find parametric equations of r and of s.

(b) Determine the relative position of r and s (parallel, incident or skew) (c) Find a vector of length 1 perpendicular to both r and s.

4) Consider the matrix M =

1 0 1 0 1 0 1 0 1

(a) Calculate the real eigenvalues of M , and their algebraic and geometric multiplicities.

(b) Determine whether M is diagonalizable.

(c) For each eigenvalue find an orthonormal basis of its eigenspace.

(2)

Name: Matriculation number:

February 6, 2020, L.A.G. exam

Solve the following exercises, explaining clearly each passage:

1) Consider the matrix M = kI + A · A

where k is a real parameter, I is the 2 × 2 identity matrix and A = 1 1 1 1



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

(b) For which values of k is M invertible?

(c) Compute the inverse of M for the values of k in (b).

2) Consider the linear transformation f : R

2

→ R

3

defined by

f ( x y

 ) = x

 1 2 1

 − y

 2 4 2

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

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

(c) Determine all the vectors v ∈ R

2

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

 2 4 2

3) In the 3-dim. euclidean space consider the line r passing through the origin, and perpendicular to the plane x + z = 400, and the line s passing through the point (1, 0, 1) and parallel to the vector (0, 500, 500).

(a) Find parametric equations of r and of s.

(b) Determine the relative position of r and s (parallel, incident or skew) (c) Find a vector of length 1 perpendicular to both r and s.

4) Consider the matrix M =

−1 1 1

0 −1 1

0 0 1

(a) Calculate the real eigenvalues of M , and their algebraic and geometric multiplicities.

(b) Determine whether M is diagonalizable.

(c) For each eigenvalue find an orthonormal basis of its eigenspace.

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