assign1 - (0 0(1 1 1(1 2 3 6 Prove that if ~x,~u,~v ∈ R n...

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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 simplified 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 = {
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Unformatted text preview: (0 , , 0) , (1 , 1 , 1) , (1 , 2 , 3) } . 6. Prove that if ~x,~u,~v ∈ R n and a, b ∈ R are scalars, then a) ( a + b ) ~x = a~x + b~x . b) ~x · ( a~v + b~u ) = a ( ~x · ~v ) + b ( ~x · ~u ). 7. Consider the statement: “If ~a · ~ b = ~a · ~ c , then ~ b = ~ c .” a) Determine, with justification, if the statement if true or false. b) If we specify ~a 6 = ~ 0, does that change the result? 8. Let ~x,~ y ∈ R n . Prove that k ~x + ~ y k 2 = k ~x k 2 + k ~ y k 2 if and only if ~x · ~ y = 0. 1...
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This note was uploaded on 05/04/2010 for the course MATH 136 taught by Professor All during the Spring '08 term at Waterloo.

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