Transcription of Linear Algebra Test 1 - Chapters 1 and 2 Practice Problems
1 Math 2331. Linear Algebra test 1 - Chapters 1 and 2. Practice Problems Use the following vectors for questions 1-7. 1 . 2 . 1 1 . 2 0 1 .. u= v= w= 2 . 4 1 1.. 2 0 2 . 1 .. 2 . 1. Find u i w 2. Find u iv 3. Find 2u + 4 w 4. Find v w 5. Are v and w orthogonal? 6. Find the length of each vector. 7. State a unit vector in the direction of u. Answer questions 8-12 for each of the following systems: x 3y = 4 x y = 1 x + 5y = 5. 2 x 4 y = 2 x+ y =5 x + 3 y = 5. 8. Sketch the ROW PICTURE of the solution to the system. 9. Sketch the COLUMN PICTURE of the solution to the system. 10. State the system as a matrix equation. 11. State the elimination matrix E that transforms the system to upper triangular. 12. Solve by elimination and back substitution.
2 X 3y = 4. 2 x 4 y = 2. x y = 1. x+ y =5. x + 5y = 5. x + 3 y = 5. State the LU and LDU factorization of each of the following matrices. 1 3 . 13. A = . 4 5 . 1 2 3 . 14. B = 2 3 0 .. 3 0 4 .. 2 1 0 0 . 1 2 1 0 . 15. C = . 0 1 2 1 .. 0 0 1 2 . Find the inverse, if it exists, of each of the following matrices. 1 1 3 . 16. A = 4 0 5 . 2 1 3 .. 4 9 . 17. M = . 2 4 . 1 4 . 18. B = . 3 12 . 2 1 0 . 19. A = 1 2 1 .. 0 1 2 .. 20. State the transpose of matrices in Problems 16-19. 21. Given the matrices in Problems 13-19, which are symmetric? 22. State the 4x4 permutation matrix P that switches row 1 and row 3. What is the inverse of P? Given the following matrices and vectors, 1 1 3 0 1 0 . 1 2 3 . A = 4 0 5 M = P = 1 0 0 . 2 1 3 4 1 7 0 0 1.
3 1 3 2 1 . 5 . Q = 2 1 b = 3 c= x = 1 . 0 5 0 4 1 .. 23. Multiply Ax =. Ab =. 24. Multiply Mc Mb 25. Multiply AM. MA. 26. Multiply QM. PQ. 27. Find the transpose of A, M and b 28. Find the transpose of PQ and MP. Solve the following systems by elimination and back substitution: 3 2 0 0 x 1 . 4 5 0 0 y 3 . 29. = . 0 0 6 5 z 4 .. 0 0 7 6 t 1 . x + 2 y + 2z = 1. 30. 4 x + 8 y + 9 z = 3. 3y + 2z = 1.
