assign1 - 2 , 3) } . 6. Prove that if x, u,v R n and a, b R...

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Math 136 Assignment 1 Due: Wednesday, Jan 13th 1. Compute each of the following. a) (1 , 3 , 4) + ( - 1 , 1 , 2) b) 3( - 1 , 1 , - 2) - 2(2 , 0 , 3). 2. Determine the distance between P (2 , 1 , 1) and Q (1 , - 1 , 1). 3. Determine which of the following pairs of vectors is orthogonal. a) (2 , 1 , 1), ( - 2 , 1 , 3). b) ( - 1 , 3 , 6), (3 , - 1 , 0). 4. Find an equation for the plane through P (3 , - 2 , 1) and parallel to x 1 - x 2 +2 x 3 = 4. 5. For each of the following sets: i) Determine if the set is linearly dependent or linearly independent. Justify. ii) Describe geometrically the span of the set and give a simpli±ed vector equation which describes it. a) S = { (1 , 1 , - 3) , ( - 1 , - 1 , 3) } . b) T = { (1 , 2 , 1) , ( - 2 , - 4 , - 2) , (2 , 1 , 2) } . c) U = { (0 , 0 , 0) , (1 , 1 , 1) , (1 ,
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Unformatted text preview: 2 , 3) } . 6. Prove that if x, u,v R n and a, b R are scalars, then a) ( a + b ) x = ax + bx . b) x ( av + bu ) = a ( x v ) + b ( x u ). 7. Consider the statement: If a b = a c , then b = c . a) Determine, with justication, if the statement if true or false. b) If we specify a = 0, does that change the result? 8. Let x, y R n . Prove that x + y 2 = x 2 + y 2 if and only if x y = 0. 1...
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