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DEPARTMENT OF MATHEMATICS NAME: Sun Yi Student Id: 42042615

FACULTY OF SCIENCE

MATH133 S211 Mathematics IB (Advanced) Assignment 3

Tutorial Group: A Due 4:00 pm 05/10 2011

Please sign the declaration below, and staple this sheet to the front of your solutions. Your assignment must be submitted at the Science Centre, E7A Level 1. Your assignment must be STAPLED, please do not put it in a plastic sleeve. PLAGIARISM Plagiarism involves using the work of another person and presenting it as one’s own. For this assignment, the following acts constitute plagiarism: a) Copying or summarizing another person’s work. b) Where there was collaborative preparatory work, submitting substantially the same final version of any material as another student. Encouraging or assisting another person to commit plagiarism is a form of improper collusion and may attract the same penalties. STATEMENT TO BE SIGNED BY STUDENT 1. I have read the definition of plagiarism that appears above. 2. In my assignment I have carefully acknowledged the source of any material which is not my own work. 3. I am aware that the penalties for plagiarism can be very severe. 4. If I have discussed the assignment with another student, I have written the solutions independently. SIGNATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Find the general solution of Ax = b with  1 2 A= 3 4 2 4 6 8 3 5 5 5 4 6 6 6  5 7 , 7 7   v w   x = x ,   y  z   2 3 b=  4 5

2. Find a matrix A such that { (1, 2, 3), (2, 3, 4) } is a basis for the column space of A and { (3, 4, 1) } is a basis for the nullspace of A. 3. Let A be an m × n matrix. Suppose the intersection of the row space of A with the nullspace of A contains only the zero vector. Prove that every vector v in Rn can be written, in exactly one way, as v = x + y, where x is in the row...