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LAHW#03

LAHW#03. • Due October 24, 2011. 2.1 Euclidean Vector Spaces. 25.

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LAHW#03

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  1. LAHW#03 •Due October 24, 2011

  2. 2.1 Euclidean Vector Spaces • 25. • Answer and explain these questions:a. Can the span of a set be the empty set?b. Can the span of a set contain one and only one vector?c. If Span(S) = Span(T), does it follow that S = T?d. If Span(S)⊆Span(T), does it follow that S⊆T?e. If S⊆T, does it follow that Span(S)⊆Span(T)?

  3. 2.1 Euclidean Vector Spaces • 37. • Set up and solve this traffic flow problem:

  4. 2.2 Lines, Planes, and Hyperplanes • 7. • Consider a plane in R3 described by the equation3x2 – 5x3 = 7. Describe this plane in the parametric form u + tv + sw.

  5. 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}.

  6. 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.

  7. 2.2 Lines, Planes, and Hyperplanes • 26. • In R2, two random lines will likely have a point in common. Is the same to be expected in R3? Answer the relation question about a line and a plane in R3.

  8. 2.2 Lines, Planes, and Hyperplanes • (7). • Establish this assertion or find a counterexample: A linear combination of linear combinations is a linear combination.(Let u1, u2, …, um be linear combinations of vectors v1, v2, …, vn. Then a linear combination of u1, u2, …, um is also a linear combination of v1, v2, …, vn.)

  9. 2.2 Lines, Planes, and Hyperplanes • (8). • Establish this assertion or find a counterexample: Let A and B be matrices. Then A ~ Bif and only ifAT ~ BT.

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