# Math 4373     Homework

## Homework

[HK] stands for Hoffman-Kunze, [H] stands for Halmos.
Fields [Due Tuesday 5th Sept]
• Verify that the complex numbers form a field.
• [H] page 2, questions 1,5,6,7.
• Prove that the set of all constructibele numbers is a field.
Vector spaces and subspaces [Due Tuesday 5th Sept]
• [H] pge 6: questions 2,4,5
• [HK] section 2.1: questions 5, 7
• [HK] section 2.2: questions 1,2,4,6,8,9
• Show that the space of n-by-n matrices is the sum of the space of
skew-symmetric matrices and the space of symmetric matrices.
Bases and dimension (Due Tuesday, Sept. 12)
• [H] section 7: questions 6, 7.
• [H] section 8: questions 1,3.
• [HK] section 2.3: questions 9, 12, 14.
Linear Transformations and Matrices; isometries of the euclidean plane (Due Tuesday, Oct. 10)
• [HK] section 3.2: questions 7, 8, 11.
• [HK] section 3.3: question 3.
• [HK] section 3.4: questions 6, 8 (definition on page 94), 9, 12.
• Write down explicit coordinate map (x,y) --> (?, ?) for the isometry consisting of rotation about the point (2,3) through Pi radians.
• Show that matrix multiplication of the (3X3)-matrices discovered in class corresponds to composition of euclidean isometries.