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LAHW#03

LAHW#03. Due October 11, 2010. 2.1 Euclidean Vector Spaces. 13.

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LAHW#03

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  1. LAHW#03 Due October 11, 2010

  2. 2.1 Euclidean Vector Spaces • 13. • Devise a test to determine wherther two lines in Rn are the same. Let the lines be described as L1 = {p + tq | t∈R} and L2 = {v + tw | t∈R}. Use your test on this special case: p = (4,2,1), q = (-1,3,2), v = (1,11,7), and w = (3,-9,-6).

  3. 2.1 Euclidean Vector Spaces • 18. • Show that if u, v, and w are three points in R2, then for suitable real numbers α, β, and γ, not all zero, we have αu+βv+γw = 0.

  4. 2.2 Lines, Planes, and Hyperplanes • 1. • Find a parametric form for the line in R2 that passes through the points (7, 3) and (-5, 6). Is the answer unique?

  5. 2.2 Lines, Planes, and Hyperplanes • 2. • In R5 does the line described parametrically by (3,4,-5,6,2)+t(2,-2,1,3,6) intersect the line represented by (17,-10,2,27,44)+t(-3,2,-5,1,4)?

  6. 2.2 Lines, Planes, and Hyperplanes • 8. • Let P be the set of all vectors X = (x1, x2, x3, x4) such that

  7. 2.2 Lines, Planes, and Hyperplanes • 10. • Is there a plane in R3 that contains the two lines described parametrically by (1,-2,3) + t(1,0,0) and (-2,5,-7) + s(4,-7,10)?

  8. 2.2 Lines, Planes, and Hyperplanes • 24. • Establish this assertion or find a counterexample: For two lines in Rn given parametrically by v + tw and x + sy to intersect, it is necessary and sufficient that x - v be in the span of {w, y}.

  9. 2.2 Lines, Planes, and Hyperplanes • 25. • Establish this assertion or find a counterexample: A necessary and sufficient condition for the line given parametrically by tu + (1 - t)v to contain the point 0 is that v be a scalar multiple of u – v.

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