MTH 211: Linear Algebra I - Quiz
University Level | Time: 1 Hour | Total Marks: 50
Section A: Short Answer Questions (2 marks each - Total 20 marks)
1. Define a vector space and give an example.
2. What does it mean for vectors to be linearly dependent?
3. Let A = [[1, 2], [3, 4]]. Find the determinant of A.
4. State two properties of matrix multiplication.
5. Find the rank of the matrix [[1, 2], [2, 4]].
6. What is a basis of a vector space?
7. Determine if the set {(1, 0), (0, 1), (1, 1)} is linearly independent in R².
8. Let T(x, y) = (x + y, 2x - y). Find the matrix representation of T.
9. What is the dimension of R?
10. Define an eigenvalue and eigenvector.
Section B: Structured Questions (3 marks each - Total 15 marks)
11. Let A = [[1, 3], [2, 4]]. Compute the inverse of A if it exists.
12. Find the characteristic polynomial of matrix B = [[2, 0], [0, 3]].
13. Diagonalize the matrix C = [[5, 0], [0, 1]].
14. Verify if the matrix D = [[1, 1], [0, 1]] is diagonalizable.
15. Prove whether the set {1, x, x²} is linearly independent in the space of polynomials of degree 2.
Section C: Application and Proof (5 marks each - Total 15 marks)
16. Change of Basis: Given the basis B = {(1, 1), (1, -1)} for R², express v = (2, 0) in the B-basis.
17. Subspaces: Let W R³ be defined by W = {(x, y, z) R³ : x + y + z = 0}. Prove that W is a subspace.
18. Null Space: Let A = [[1, 2, 1], [2, 4, 2]]. Find a basis for the null space of A.
University Level | Time: 1 Hour | Total Marks: 50
Section A: Short Answer Questions (2 marks each - Total 20 marks)
1. Define a vector space and give an example.
2. What does it mean for vectors to be linearly dependent?
3. Let A = [[1, 2], [3, 4]]. Find the determinant of A.
4. State two properties of matrix multiplication.
5. Find the rank of the matrix [[1, 2], [2, 4]].
6. What is a basis of a vector space?
7. Determine if the set {(1, 0), (0, 1), (1, 1)} is linearly independent in R².
8. Let T(x, y) = (x + y, 2x - y). Find the matrix representation of T.
9. What is the dimension of R?
10. Define an eigenvalue and eigenvector.
Section B: Structured Questions (3 marks each - Total 15 marks)
11. Let A = [[1, 3], [2, 4]]. Compute the inverse of A if it exists.
12. Find the characteristic polynomial of matrix B = [[2, 0], [0, 3]].
13. Diagonalize the matrix C = [[5, 0], [0, 1]].
14. Verify if the matrix D = [[1, 1], [0, 1]] is diagonalizable.
15. Prove whether the set {1, x, x²} is linearly independent in the space of polynomials of degree 2.
Section C: Application and Proof (5 marks each - Total 15 marks)
16. Change of Basis: Given the basis B = {(1, 1), (1, -1)} for R², express v = (2, 0) in the B-basis.
17. Subspaces: Let W R³ be defined by W = {(x, y, z) R³ : x + y + z = 0}. Prove that W is a subspace.
18. Null Space: Let A = [[1, 2, 1], [2, 4, 2]]. Find a basis for the null space of A.
