TRENTO, A.A. 2019/20
MATHEMATICS FOR DATA SCIENCE/BIOSTATISTICS EXERCISE SHEET # 11
Important! In solving the exercises
• explain what you are doing,
• explain why you are doing what you are doing, and
• spell out all intermediate steps.
Notice! For technical reasons, sometimes in the following exercises we write something like “e
1+ ( −1)e
2”, which just means e
1− e
2.
Exercise 11.1. Define eigenvalues and eigenvectors.
Show that if v is an eigenvector for the linear function f with respect to the eigenvalue λ, and a ̸= 0 is a number, then av is also an eigenvector for f with respect to the eigenvalue λ.
Exercise 11.2. Let V be a vector space of dimension 2, and let e
1, e
2be a base.
Consider the linear function f : V → V given by {
f (e
1) = 5e
1+ 2e
2f (e
2) = ( −12)e
1+ ( −5)e
2Write the matrix A of f with respect to the basee
1, e
2. Compute the characteristic polynomial (in the variable λ)
det(A − λI),
where I is the 2 ×2 identity matrix. Find the eigenvalues λ
1, λ
2if A as the solutions to the equation
det(A − λI) = 0.
For each eigenvalue λ
i, find an eigenvector g
i. Show that g
1, g
2is a base of V , and write the matrix B of f with respect to the base g
1, g
2.
Exercise 11.3. Let V be a vector space of dimension 2, and let e
1, e
2be a base.
Consider the linear function f : V → V given by {
f (e
1) = ( −4)e
1+ 10e
2f (e
2) = ( −3)e
1+ 7e
2Write the matrix A of f with respect to the basee
1, e
2. Compute the characteristic polynomial (in the variable λ)
det(A − λI),
where I is the 2 ×2 identity matrix. Find the eigenvalues λ
1, λ
2if A as the solutions to the equation
det(A − λI) = 0.
For each eigenvalue λ
i, find an eigenvector g
i. Show that g
1, g
2is a base of V , and
write the matrix B of f with respect to the base g
1, g
2.
2 EXERCISE SHEET # 11