sample mid2

Xy 1 1 5 let p be the plane in r3 with vector equation

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Unformatted text preview: and ￿￿ ￿. x y b) Determine the value of ￿ · ￿ . xy 1 1 5. Let P be the plane in R3 with vector equation ￿ = s 4 + t 2. x 1 0 a) Find a scalar equation for P of the form ax1 + bx2 + cx3 = d. 1 b) Find the projection of ￿ = 0 onto P . x 0 1 3 1 5 1 , −1 , 1 represents a line or a plane 6. Determine if the span of 0 2 −1 in R3 and give a simplified vector equation which describes it. 7. Determine, with proof, which of the following are subspaces of R3 and which are not. If the set is a subspace, then find a basis for it. (a) The solution set of the system of equations 3 x1 + 4 x2 − x3 = 5 2 x1 − 3 x2 + 2 x3 = 6 x1 (b) The set S = x2 ∈ R3 | x1 = x2 − x3 . x3 8. Prove that if ￿1 , ￿2 ∈ Rn and s, t are non-zero real numbers, then vv span{￿1 , ￿2 } = span{￿1 , s￿1 + t￿2 } vv vv v 9. Let S = {￿1 , . . . , ￿k } be a set in Rn . v v a) Prove that if S is linearly independent, then k ≤ n. b) Prove that if span S = Rn , then k ≥ n. 10. Let B =...
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This note was uploaded on 01/16/2014 for the course MATH 136 taught by Professor All during the Winter '08 term at Waterloo.

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