MIT18_06S10_exam2_s10_soln

MIT18_06S10_exam2_s10_soln - 18.06 Quiz 2 April 7 2010...

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18.06 Quiz 2 April 7, 2010 Professor Strang Your PRINTED name is: 1. Your recitation number or instructor is 2. 3. 1. (33 points) (a) Find the matrix P that projects every vector b in R 3 onto the line in the direction of a = (2 , 1 , 3) . Solution The general formula for the orthogonal projection onto the column space of a matrix A is P = A ( A T A ) 1 A T . Here, so that P = 1 14 4 2 2 1 6 3 6 3 9 2 A = 1 3 Remarks: Since we’re projecting onto a one-dimensional space, A T A is just a number and we can write things like P = ( AA T ) / ( A T A ). This won’t work in general. You don’t have to know the formula to do this. The i th column of P is, pretty much by definition, the projection of e i ( e 1 = (1 , 0 , 0), e 2 = (0 , 1 , 0), e 3 = (0 , 0 , 1)) onto the line in the direction of a . And this is something you should know how to do without a formula. RUBRIC: There was some leniency for computational errors, but otherwise there weren’t many opportunities for partial credit.
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(b) What are the column space and nullspace of P ? Describe them geometrically and also give a basis for each space. Solution The column space is the line in R 3 in the direction of a = (2 , 1 , 3). One basis for it is 2 1 3 and there’s not really much choice in giving this basis (you can rescale by a non-zero con- stant). The nullspace is the plane in R 3 that is perpendicular to a = (2 , 1 , 3) (i.e., 2 x + y + z = 0.) One basis for it is 3 1 0 , 2 2 0 though there are a lot of different looking choices for it (any two vectors that are perpendic- ular to a and not in the same line will work). RUBRIC: 6 points for giving a correct basis, and 4 points for giving the complete geometric description. Note that it is not correct to say e.g., N ( P ) = R 2 . It is correct to say that N ( P ) is a (2-dimensional) plane in R 3 , but this is not a complete geometric description unless you say (geometrically) which plane it is: the one perpendicular to a /to the line through a .
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(c) What are all the eigenvectors of P and their corresponding eigenvalues? (You can use the geometry of projections, not a messy calculation.) The diagonal entries of P add up to . Solution The diagonal entries of P add up to 1 = the sum of the eigenvalues Since P is a projection, it’s only possible eigenvalues are λ = 0 (with multiplicity equal to the dimension of the nullspace, here 2) and λ = 1 (with multiplicity equal to the dimension of the column space, here 1). So, a complete list of
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