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Midterm study guide On the midterm, there will be four questions whose types are from the following list. 1. Solve a system of linear equations. 2. Compute the reduced row echelon form of a matrix. 3. Determine whether a set of vectors spans Rn or not. 4. Determine whether a set of vectors is linearly independent or not. 5. Find a linear transformation satisfying a condition. 6. Determine whether a function is a linear transformation or not. 7. Construct a matrix satisfying a condition. 8. Determine whether a matrix is invertible or not. 9. Compute the determinant of a matrix. 10. Find the volume of a region. 11. Determine whether a set with an addition and a scalar multiplication is a vector space or not. 12. Determine whether a subset of a vector space is a subspace or not. 13. Find a basis of a vector space (e.g. Col A, Nul A, Col A ∩ Col B, Nul A ∩ Nul B, Ker T , Im T , Pn , · · ·.) 14. Find a basis of a vector space satisfying some property. 15. Find the dimension of a vector space. 16. Determine whether a set in a vector space is linearly independent or not. 17. Determine whether a set in a vector space spans the vector space or not. 18. Let B be a basis of a vector space V , and let x ∈ V be an element. Find [x]B . 19. Let A be a m × n matrix. Given dim Col A, find dim Row A, dim Nul A, or dim Nul AT . 20. Let B and C be bases of a vector space V . Find the change-of-coordinates matrix from B to C. 21. Let A be an n × n matrix. Find its eigenvalues and eigenvectors. 22. Let A be an n × n matrix. Find its characteristic equation. 1 23. Let A be an n × n matrix. Determine whether it is diagonalizable or not. When it is diagonalizable, find a diagonal matrix D and an invertible matrix P such that A = P DP −1 . 24. Let T : V → V be a linear transformation where V is a vector space. Compute [T ]B . 25. Let T : V → V be a linear transformation where V is a vector space. Find a basis B of V such that the matrix [T ]B is a diagonal matrix. 26. Determine whether a subset of Rn is orthogonal or not. 27. Let y be an element of Rn , and let W be a subspace of Rn . Find the orthogonal projection of y onto W . 28. Find an orthonormal basis of a vector space. 29. Let A be an m × n matrix, and let b ∈ Rn be an element. Find a leastsquare solution of Ax = b. 30. Let V be a vector space. Determine whether a function that associates hu, vi to each pair u and v in V is an inner product or not. 31. Let A be a symmetric n × n matrix. Orthogonally diagonalize it. 32. Let A be a symmetric n × n matrix. Determine whether the quadratic form xT Ax is positive definite or not. 2