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1) Compute the rank of the matrix

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Name: Matriculation number:

September 2, 2019, L.A.G. exam

Solve the following exercises, explaining clearly each passage:

1) Compute the rank of the matrix

1 1 k

0 k − 1 0

k − 1 0 0

 depending on the pa- rameter k. For which values of k is the matrix invertible?

2) Consider the linear transformation f : R

3

→ R

2

defined by multiplication by the matrix A = 1 2 −1

2 4 −2



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

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

(c) Determine all the vectors v ∈ R

3

, if they exist, such that f (v) = 10 20



3) In the euclidean plane consider the line r passing through the point P = (1, −1), parallel to the vector v = (1, 2), and the line s, with cartesian equation x + y = 1

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

(b) Determine the intersection point of r and s.

(c) Determine the angle between r and s.

(d) Determine the distance between the line r and the origin.

4) Consider the matrix M =

1 0 0 1 1 1 1 0 2

(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.

(d) Is there an orthogonal matrix C such that C

T

M C is a diagonal matrix?

(2)

Name: Matriculation number:

September 2, 2019, L.A.G. exam

Solve the following exercises, explaining clearly each passage:

1) Compute the rank of the matrix

1 k 1

1 0 k − 1 k − 2 0 0

depending on the parameter k. For which values of k is the matrix invertible?

2) Consider the linear transformation f : R

2

→ R

3

defined by multiplication by the matrix A =

1 1

0 2

−1 0

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

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

(c) Determine all the vectors v ∈ R

2

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

 0 1 0

3) In the euclidean plane consider the line r passing through the point P = (1, −1), parallel to the vector v = (2, 1), and the line s, with cartesian equation x − y = 1

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

(b) Determine the intersection point of r and s.

(c) Determine the angle between r and s.

(d) Determine the distance between the line r and the origin.

4) Consider the matrix M =

2 0 0

1 1 1

−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.

(d) Is there an orthogonal matrix C such that C

T

M C is a diagonal matrix?

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