Assignment 1

Assignment 1 - =-1-5 x 1 +15 x 2-30 x 3 +2 x 4 +8 x 5 = 2 2...

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MATH 223, Linear Algebra Winter, 2008 Assignment 1, due in class January 16, 2008 1. Let z = 4 - 2 i and w = 8 + i . Find ¯ z , ¯ w , z + w , z - w , z · w and z w (all in the form a + bi with a and b real numbers). Find the absolute value of each of these 6 numbers. 2. Show that if z and w are any two complex numbers, then z · w = ¯ z · ¯ w . Use this to show that if A and B are any complex matrices, then A · B = ¯ A · ¯ B . [N.B. The conjugate ¯ A of a matrix A is the most obvious thing — you just replace each entry of A by its conjugate. Also, we of course assume here that A · B is defined.] 3. Solve each of the following systems of equations. That is, find the unique solution if there is one, the general solution in vector parametric form if there is more than one solution, or explain why there is no solution if that is the case. Use augmented matrices. (a) This one’s over the field R , the reals. x 1 - 3 x 2 +7 x 3 +5 x 5 = 8 - 5 x 1 +15 x 2 - 30 x 3 +2 x 4 +8 x 5 = 0 2 x 1 - 6 x 2 +9 x 3 - 2 x 4 - 23 x 5 = 10 (b) This one’s also over the field R . x 1 - 3 x 2 +7 x 3 +5 x 5
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Unformatted text preview: =-1-5 x 1 +15 x 2-30 x 3 +2 x 4 +8 x 5 = 2 2 x 1-6 x 2 +9 x 3-2 x 4-23 x 5 = 1 (c) This ones over the eld C , the complex numbers. x 1 +(-2 + 2 i ) x 2 +(7-i ) x 3 = 10 + i (3-i ) x 1 +(-2 + 10 i ) x 2 +(24-6 i ) x 3 = 31-15 i 4 ix 2 +4 x 3 = 8-8 i (d) This ones over the two-element eld Z 2 . x 1 + x 3 + x 4 = 0 x 1 + x 2 + x 4 + x 5 = 1 x 1 + x 3 + x 5 = 0 x 1 + x 2 + x 5 = 1 In this case, explicitly list all the solutions. 4. (a) Let A = a b c d be any 2 2 matrix. Show that there is a nonzero vector ~v with A~v = ~ 0 if and only if ad-bc = 0. (b) Find all complex numbers (if any) such that -1 2-2-1 ~v = ~v has a nonzero solution ~v . (c) For each you found in the previous part, nd all vectors ~v such that -1 2-2-1 ~v = ~v . 1...
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This note was uploaded on 04/29/2008 for the course MATH 223 taught by Professor Loveys during the Fall '07 term at McGill.

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