18.06 Problem Set 1Due Wednesday, 10 September 2008 at 4 pm in 2-106.Problem 1: Do problem 25 from section 1.2 in the book.Problem 2: Do problem 29 from section 1.2.Problem 3: Do problem 5 from section 2.1.Problem 4: Do problem 13 from section 2.2.Problem 5: Do problem 26 from section 2.2.Problem 6: Do problem 30 from section 2.3.Problem 7: Do problem 35 in section 2.4Problem 8:1. In three dimensions the set of all vectors perpendicular to (1,1,1) form 1) aline 2) a plane 3) a point 4) all of 3d.2. Draw a cube with eight vertices (±1 , ±1, ±1) There are twelve edges eachwith a midpoint. How many of the vectors from the origin to the midpointsare perpendicular to (1,1,1)?3. Investigate the cosine of angles of “consecutive” midpoints (your job is tofigure out what consecutive here must mean!)4. Optional: What figure is formed from these midpoints?1Problem 9: In your computational environment set up a 4 x 4 matrix A of yourown choice. In succession reset the upper left element of A to be 1, 2, 3, 4, etcand compute the determinant. (Many packages have commands such as det(A) orDet[A] (Mathematica) or Determinant(A) (Maple[LinearAlgebra])). For example,A(1,1)=1, det(A), A(1,1)=2, det(A), A(1,1)=3, det(A), etc. What can you sayabout the succesive determinants?Now compute the upper left element of inv(A)*det(A) (inverse times determi-nant, which may have slightly different names in different languages). What canyou say about the sequence of upper left elements as we run through 1,2,3,4 inthe upper left of A? For example, A(1,1)=1, B=inv(A)*det(A), B(1,1), A(1,1)=2,B=inv(A)*det(A), B(1,1), A(1,1)=3, B=inv(A)*det(A);B(1,1).N.B. If you run into a matrix that has no inverse (or is computationally near amatrix with no inverse), you might have difficulties: large numbers, infs, or NaNs(”Not a Number”) depending on the package. Probably easiest to change one num-ber in your matrix and these difficulties should
View Full Document
Unlocking...