Find the matrix of the mapping (x, y) 7 → (x + y, y) . Using any method you prefer, find alleigenvectors of the mapping and the corresponding eigenvalues.

Sure, let's start by finding the matrix of the mapping (x, y) 7 → (x + y, y).

The given mapping can be written as:

x' = x + y y' = y

We can write this in matrix form as:

[1 1] [0 1]

This is the matrix of the mapping.

Next, let's find the eigenvalues. The eigenvalues of a matrix A are the roots of the characteristic equation, which is given by det(A - λI) = 0, where I is the identity matrix and λ are the eigenvalues.

For our matrix, we have:

det([1-λ 1] = 0 [0 1-λ])

Solving this gives us the characteristic equation (1-λ)^2 = 0. The roots of this equation are the eigenvalues of the matrix. In this case, we have one eigenvalue of λ = 1 with multiplicity 2.

Finally, let's find the eigenvectors. The eigenvectors are the null space of the matrix (A - λI). For λ = 1, we have:

[A - λI]v = 0 [0 1] * [v1] = 0 [0 0] [v2]

Solving this system gives us v1 = 0 and v2 is free. Therefore, the eigenvectors are of the form (0, v2) for any real number v2.