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Assignment 1 - MATH 223 Linear Algebra Winter 2010...

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MATH 223, Linear Algebra Winter, 2010 Assignment 1, due in class January 18, 2010 1. Let z = 5 + 3 i and w = 4 - 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. (a) Prove that for any complex numbers z 1 and z 2 , z 1 + z 2 = z 1 + z 2 and z 1 · z 2 = z 1 · z 2 You may use any properties about multiplication and addition of real numbers. (b) If A is a matrix over the complex numbers, we let ¯ A be the most obvious thing — it is obtained from A by replacing each entry by its conjugate. Supposing that A · B is defined, show that A · B = ¯ A · ¯ B . 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.
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