Download Midterm study guide

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.
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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.
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