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Given the straight lines and

find the right statement.

The lines are skew.

is orthogonal to the plane .

The lines are coplanar.

is orthogonal to the plane .

. Consider the subspace . Find the right statement:

Let , , be in

, that

is the vector space of polynomials in

having real coefficients and degree

less of equal than

Consider the vectors in 3-space such that , then it is necessarily true that

Let . Which one of the following

statements is true?

The orthogonal projection of on is .

The orthogonal projection of on is .

The orthogonal projection of on is .

The orthogonal projection of on is .

and the plane of equations respectively

In the euclidean 3-dimensional space with a fixed cartesian system, let

us consider the sphere

Which of the following statements is true?

The distance of the center of from the plane is equal to .

is tangent to .

The radius of the circumference is equal to .

Let be a matrix such that

and let be endormorphism associated to with respect to some base of .

Find the correct statement.

The characteristic polynomial of the matrix is divisible by .

The linear system has solutions for any

The matrix has rank .

The matrix has non-zero determinant.

is not a finitely dimensional vector space.

Let be the space of functions of the real variable which are differentiable infinitely many times.

Consider the linear map such that where is the first derivative of with respect to .

Find the right statement.

is an eigenvalue of .

does not have real eigenvalues.

is injective.

Let be a linear map such that

for some .

It is necessarily true that:

is empty.

is an eigenvector of .

is the only eigenvalue of .

if is odd, then has at least a real eigenvalue.

Let and consider the matrix

Find the true statement.

for finitely many values of .

is invertible for all values of .

There exists a value of such that .

for two values of .