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Unformatted text preview: singular? (Hint: It is helpful to notice that there is one obvious choice of x that would make the columns of A dependent.) 4. (12 points) Consider the vectors ~v 1 = 1 1 1 1 , ~v 2 = 1 1 , ~v 3 = 11 (a) Compute the volume of the 3dimensional parallelogram in R 4 with edges ~v 1 , ~v 2 , ~v 3 . (b) Find the vector in V = span( ~v 1 , ~v 2 , ~v 3 ) that is closest to ~ b = 1 2 3 4 . 5. (12 points) True or False: As usual, justify your answers! (a) If A is a square matrix then A and A T have the same eigenvalues. (b) If det A = 1 then A is an orthogonal matrix. (c) If A is an orthogonal matrix, and a ij is any entry of the matrix A then1 ≤ a ij ≤ 1. (d) Suppose that A and B are n × n matrices, that det A = 3 and that det B = 6. Then det((2 A ) T B1 ) = 1....
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 Spring '08
 Staff
 Math, Linear Algebra, Algebra, The Matrix Reloaded, The Matrix Revolutions, Orthogonal matrix

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