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Linear Algebra and Geometry
September 2017
Instructions:
- Write name, surname, student number in BLOCK LETTERS in the spaces provided.
- For each of the quiz of the first part, check the right answer in the table on this page.
- Write the answers to the questions of the exercises of the second part on the blank pages at the end of each exercise. You must explain your answers.
SURNAME, NAME:
IDENTIFICATION NUMBER:
PROFESSOR:
Q1 a (^) b c (^) d Q5 a (^) b c (^) d
Q2 a (^) b c (^) d Q6 a (^) b c (^) d
Q3 a^ b c^ d Q7 a^ b c^ d
Q4 a (^) b c (^) d Q8 a (^) b c (^) d
Do not write here
QUIZ EXERCISE TOTAL
QUIZ
Q1. Consider the matrix
A =
( 0 0 0 0
) .
Find the correct statement. (a) A is a diagonalizable matrix. (b) x^2 + 1 is the characteristic polynomial of A. (c) A is an invertible matrix. (d) A has the eigenvalue 1.
Q2. Consider the linear system of equation S : AX = B where
A =
( 1 1 − 1 1
) , B =
( 0 0
) .
Find the correct statement. (a) S has exactly one solution. (b) S does not have solutions. (c) S has ∞^1 solutions. (d) S has ∞^2 solutions.
Q3. Consider the matrix
A =
Find the correct statement. (a) The determinant of A is det(A) = 1. (b) The rank of A is ρ(A) = 1. (c) The determinant of A is det(A) = 2. (d) The rank of A is ρ(A) = 2.
Q4. Consider the linear map f : R^2 → R^2 defined as f (x, y) = (x, x + y).
Find the correct statement. (a) f is an isomorphism. (b) f is not surjective. (c) Im(f ) = ∅. (d) (1, 1) ∈ Ker(f ).
Q5. Given the matrix
A =
Find the correct statement. (a) A is invertible. (b) The determinant of A is det(A) = 1. (c) λ = 1 is an eigenvalue of A. (d) x^2 − 2 x is the characteristic polynomial of A.
EXERCISES
Exercise 1. Let t ∈ R. Consider the linear map f : R^3 → R^3 with associated matrix in canonical bases
A =
1 3 0 3 2 0 0 0 t^2 − 1
.
(i) For t = 2 prove that f is an isomorphism.
(ii) For t = − 1 find a basis for Ker(f ).
Solution of exercise 1:
Solution of exercise 1:
