06-240/Term Test

From Drorbn
Jump to navigationJump to search

The Test

Front Page

Do not turn this page until instructed.

Math 240 Algebra I - Term Test


University of Toronto, October 24, 2006

Solve the 5 problems on the other side of this page.

Each of the problems is worth 20 points.

You have an hour and 45 minutes.

Notes.

  • No outside material other than stationary and a basic calculator is allowed.
  • We will have an extra hour of class time in our regular class room on Thursday, replacing the first tutorial hour.
  • The final exam date was posted by the faculty - it will take place on Wednesday December 13 from 2PM until 5PM at room 3 of the Clara Benson Building, 320 Huron Street (south west of Harbord cross Huron, home of the Faculty of Physical Education and Health).
Good Luck!

Questions Page

Solve the following 5 problems. Each of the problems is worth 20 points. You have an hour and 45 minutes.

Problem 1. Let be a field with zero element , let be a vector space with zero element and let be some vector. Using only the axioms of fields and vector spaces, prove that .

Problem 2.

  1. In the field {\mathbb C} of complex numbers, compute
         and     .
  2. Working in the field of integers modulo 7, make a table showing the values of for every .

Problem 3. Let be a vector space and let and be subspaces of . Prove that is a subspace of iff or .

Problem 4. In the vector space , decide if the matrix is a linear combination of the elements of .

Problem 5. Let be a finite dimensional vector space and let and be subspaces of for which . Denote the linear span of by . Prove that .

Good Luck!

Solution Set

Students are most welcome to post a solution set here.

1. (by F3)

(by VS8)

By VS4,

Add to both sides of

(by construction)

(by VS3)

2. 1)

2)